Daily Papers

We adapt previous research on category theory and topological unsupervised learning to develop a functorial perspective on manifold learning, also known as nonlinear dimensionality reduction. We first characterize manifold learning algorithms as functors that map pseudometric spaces to optimization objectives and that factor through hierarchical clustering functors. We then use this characterization to prove refinement bounds on manifold learning loss functions and construct a hierarchy of manifold learning algorithms based on their equivariants. We express several popular manifold learning algorithms as functors at different levels of this hierarchy, including Metric Multidimensional Scaling, IsoMap, and UMAP. Next, we use interleaving distance to study the stability of a broad class of manifold learning algorithms. We present bounds on how closely the embeddings these algorithms produce from noisy data approximate the embeddings they would learn from noiseless data. Finally, we use our framework to derive a set of novel manifold learning algorithms, which we experimentally demonstrate are competitive with the state of the art.

Given a complex of groups, we construct a new class of complex of groups that records its local data and offer a functorial perspective on the statement that complexes of groups are locally developable. We also construct a new notion of an immersion of complexes of groups and establish that a locally isometric immersion of a complex of groups into a non-positively curved complex of groups is pi_1-injective. Furthermore, the domain complex of groups is developable and the induced map on geometric realizations of developments is an isometric embedding.

Optics and lenses are abstract categorical gadgets that model systems with bidirectional data flow. In this paper we observe that the denotational definition of optics - identifying two optics as equivalent by observing their behaviour from the outside - is not suitable for operational, software oriented approaches where optics are not merely observed, but built with their internal setups in mind. We identify operational differences between denotationally isomorphic categories of cartesian optics and lenses: their different composition rule and corresponding space-time tradeoffs, positioning them at two opposite ends of a spectrum. With these motivations we lift the existing categorical constructions and their relationships to the 2-categorical level, showing that the relevant operational concerns become visible. We define the 2-category 2-Optic(C) whose 2-cells explicitly track optics' internal configuration. We show that the 1-category Optic(C) arises by locally quotienting out the connected components of this 2-category. We show that the embedding of lenses into cartesian optics gets weakened from a functor to an oplax functor whose oplaxator now detects the different composition rule. We determine the difficulties in showing this functor forms a part of an adjunction in any of the standard 2-categories. We establish a conjecture that the well-known isomorphism between cartesian lenses and optics arises out of the lax 2-adjunction between their double-categorical counterparts. In addition to presenting new research, this paper is also meant to be an accessible introduction to the topic.

Many clustering schemes are defined by optimizing an objective function defined on the partitions of the underlying set of a finite metric space. In this paper, we construct a framework for studying what happens when we instead impose various structural conditions on the clustering schemes, under the general heading of functoriality. Functoriality refers to the idea that one should be able to compare the results of clustering algorithms as one varies the data set, for example by adding points or by applying functions to it. We show that within this framework, one can prove a theorems analogous to one of J. Kleinberg, in which for example one obtains an existence and uniqueness theorem instead of a non-existence result. We obtain a full classification of all clustering schemes satisfying a condition we refer to as excisiveness. The classification can be changed by varying the notion of maps of finite metric spaces. The conditions occur naturally when one considers clustering as the statistical version of the geometric notion of connected components. By varying the degree of functoriality that one requires from the schemes it is possible to construct richer families of clustering schemes that exhibit sensitivity to density.

Lenses are a well-established structure for modelling bidirectional transformations, such as the interactions between a database and a view of it. Lenses may be symmetric or asymmetric, and may be composed, forming the morphisms of a monoidal category. More recently, the notion of a learner has been proposed: these provide a compositional way of modelling supervised learning algorithms, and again form the morphisms of a monoidal category. In this paper, we show that the two concepts are tightly linked. We show both that there is a faithful, identity-on-objects symmetric monoidal functor embedding a category of asymmetric lenses into the category of learners, and furthermore there is such a functor embedding the category of learners into a category of symmetric lenses.

We apply category theory to extract multimodal document structure which leads us to develop information theoretic measures, content summarization and extension, and self-supervised improvement of large pretrained models. We first develop a mathematical representation of a document as a category of question-answer pairs. Second, we develop an orthogonalization procedure to divide the information contained in one or more documents into non-overlapping pieces. The structures extracted in the first and second steps lead us to develop methods to measure and enumerate the information contained in a document. We also build on those steps to develop new summarization techniques, as well as to develop a solution to a new problem viz. exegesis resulting in an extension of the original document. Our question-answer pair methodology enables a novel rate distortion analysis of summarization techniques. We implement our techniques using large pretrained models, and we propose a multimodal extension of our overall mathematical framework. Finally, we develop a novel self-supervised method using RLVR to improve large pretrained models using consistency constraints such as composability and closure under certain operations that stem naturally from our category theoretic framework.

We introduce a Heegaard-Floer homology functor from the category of oriented links in closed 3-manifolds and oriented surface cobordisms in 4-manifolds connecting them to the category of F[v]-modules and F[v]-homomorphisms between them, where F is the field with two elements. In comparison with previously defined TQFTs for decorated links and link cobordisms, the construction of this paper has the advantage of being independent from the decoration. Some of the basic properties of this functor are also explored.

This book provides an inviting tour through sheaf theory, from the perspective of applied category theory and pitched at a less specialized audience than is typical with introductions to sheaves. The book makes it as easy as possible for the reader new to sheaves, by motivating and developing the theory via a broad range of concrete examples and explicit constructions, including applications to n-colorings of graphs, satellite data, chess problems, Bayes nets, musical performance, complexes, and more. Included is an extended first chapter introducing and motivating all the necessary category-theoretical background, again with a strong emphasis on concrete examples. A new and unabridged version (including a fifth chapter on more advanced topics and a conclusion) will be available with MIT Press.

We present a novel application of category theory for deep learning. We show how category theory can be used to understand and work with the linear layer functions of group equivariant neural networks whose layers are some tensor power space of R^{n} for the groups S_n, O(n), Sp(n), and SO(n). By using category theoretic constructions, we build a richer structure that is not seen in the original formulation of these neural networks, leading to new insights. In particular, we outline the development of an algorithm for quickly computing the result of a vector that is passed through an equivariant, linear layer for each group in question. The success of our approach suggests that category theory could be beneficial for other areas of deep learning.

In an unpublished preprint batanin, Batanin conjectures that it is possible to take `slices' of a globular operad, thereby isolating the algebraic structure in each dimension. It was further hypothesised that the slices of a globular operad for some theory of higher category contain essential information about those higher categories, namely whether or not they are equivalent to the fully weak variety. In this paper, we use the theory of presentations for globular operads developed in Me to provide a concrete definition of slices, and calculate the slices for several key theories of n-category.

We define the bicategory of Graph Convolutional Neural Networks GCNN_n for an arbitrary graph with n nodes. We show it can be factored through the already existing categorical constructions for deep learning called Para and Lens with the base category set to the CoKleisli category of the product comonad. We prove that there exists an injective-on-objects, faithful 2-functor GCNN_n to Para(CoKl(R^{n times n} times -)). We show that this construction allows us to treat the adjacency matrix of a GCNN as a global parameter instead of a a local, layer-wise one. This gives us a high-level categorical characterisation of a particular kind of inductive bias GCNNs possess. Lastly, we hypothesize about possible generalisations of GCNNs to general message-passing graph neural networks, connections to equivariant learning, and the (lack of) functoriality of activation functions.

Enhancing the intelligibility and interpretability of machine learning is a crucial task in responding to the demand for Explicability as an AI principle, and in promoting the better social implementation of AI. The aim of our research is to contribute to this improvement by reformulating machine learning models through the lens of category theory, thereby developing a semantic framework for structuring and understanding AI systems. Our categorical modeling in this paper clarifies and formalizes the structural interplay between residuals and parameters in supervised learning. The present paper focuses on the multiple linear regression model, which represents the most basic form of supervised learning. By defining two concrete categories corresponding to parameters and data, along with an adjoint pair of functors between them, we introduce our categorical formulation of supervised learning. We show that the essential structure of this framework is captured by what we call the Gauss-Markov Adjunction. Within this setting, the dual flow of information can be explicitly described as a correspondence between variations in parameters and residuals. The ordinary least squares estimator for the parameters and the minimum residual are related via the preservation of limits by the right adjoint functor. Furthermore, we position this formulation as an instance of extended denotational semantics for supervised learning, and propose applying a semantic perspective developed in theoretical computer science as a formal foundation for Explicability in AI.

This thesis introduces quantum natural language processing (QNLP) models based on a simple yet powerful analogy between computational linguistics and quantum mechanics: grammar as entanglement. The grammatical structure of text and sentences connects the meaning of words in the same way that entanglement structure connects the states of quantum systems. Category theory allows to make this language-to-qubit analogy formal: it is a monoidal functor from grammar to vector spaces. We turn this abstract analogy into a concrete algorithm that translates the grammatical structure onto the architecture of parameterised quantum circuits. We then use a hybrid classical-quantum algorithm to train the model so that evaluating the circuits computes the meaning of sentences in data-driven tasks. The implementation of QNLP models motivated the development of DisCoPy (Distributional Compositional Python), the toolkit for applied category theory of which the first chapter gives a comprehensive overview. String diagrams are the core data structure of DisCoPy, they allow to reason about computation at a high level of abstraction. We show how they can encode both grammatical structures and quantum circuits, but also logical formulae, neural networks or arbitrary Python code. Monoidal functors allow to translate these abstract diagrams into concrete computation, interfacing with optimised task-specific libraries. The second chapter uses DisCopy to implement QNLP models as parameterised functors from grammar to quantum circuits. It gives a first proof-of-concept for the more general concept of functorial learning: generalising machine learning from functions to functors by learning from diagram-like data. In order to learn optimal functor parameters via gradient descent, we introduce the notion of diagrammatic differentiation: a graphical calculus for computing the gradients of parameterised diagrams.

We introduce the category of information structures, whose objects are suitable diagrams of measurable sets that encode the possible outputs of a given family of observables and their mutual relationships of refinement; they serve as mathematical models of contextuality in classical and quantum settings. Each information structure can be regarded as a ringed site with trivial topology; the structure ring is generated by the observables themselves and its multiplication corresponds to joint measurement. We extend Baudot and Bennequin's definition of information cohomology to this setting, as a derived functor in the category of modules over the structure ring, and show explicitly that the bar construction gives a projective resolution in that category, recovering in this way the cochain complexes previously considered in the literature. Finally, we study the particular case of a one-parameter family of coefficients made of functions of probability distributions. The only 1-cocycles are Shannon entropy or Tsallis alpha-entropy, depending on the value of the parameter.

In "Backprop as functor", the authors show that the fundamental elements of deep learning -- gradient descent and backpropagation -- can be conceptualized as a strong monoidal functor Para(Euc)toLearn from the category of parameterized Euclidean spaces to that of learners, a category developed explicitly to capture parameter update and backpropagation. It was soon realized that there is an isomorphism LearncongPara(Slens), where Slens is the symmetric monoidal category of simple lenses as used in functional programming. In this note, we observe that Slens is a full subcategory of Poly, the category of polynomial functors in one variable, via the functor Amapsto Ay^A. Using the fact that (Poly,otimes) is monoidal closed, we show that a map Ato B in Para(Slens) has a natural interpretation in terms of dynamical systems (more precisely, generalized Moore machines) whose interface is the internal-hom type [Ay^A,By^B]. Finally, we review the fact that the category p-Coalg of dynamical systems on any p in Poly forms a topos, and consider the logical propositions that can be stated in its internal language. We give gradient descent as an example, and we conclude by discussing some directions for future work.

Bayes' rule tells us how to invert a causal process in order to update our beliefs in light of new evidence. If the process is believed to have a complex compositional structure, we may observe that the inversion of the whole can be computed piecewise in terms of the component processes. We study the structure of this compositional rule, noting that it relates to the lens pattern in functional programming. Working in a suitably general axiomatic presentation of a category of Markov kernels, we see how we can think of Bayesian inversion as a particular instance of a state-dependent morphism in a fibred category. We discuss the compositional nature of this, formulated as a functor on the underlying category and explore how this can used for a more type-driven approach to statistical inference.

We derive sufficient conditions for exact functors on locally finite abelian categories to preserve Loewy diagrams of objects. We apply our results to determine sufficient conditions for induction functors associated to simple current extensions of vertex algebras to preserve Loewy diagrams.

Inspired by the foundational works by Spivak and Fong and Cruttwell et al., we introduce a categorical framework to formalize Bayesian inference and learning. The two key ideas at play here are the notions of Bayesian inversions and the functor GL as constructed by Cruttwell et al.. In this context, we find that Bayesian learning is the simplest case of the learning paradigm. We then obtain categorical formulations of batch and sequential Bayes updates while also verifying that the two coincide in a specific example.

Transdisciplinary approaches are increasingly essential for addressing grand societal challenges, particularly in complex domains such as Artificial Intelligence (AI), urban planning, and social sciences. However, effectively validating and integrating knowledge across distinct epistemic and ontological perspectives poses significant difficulties. This article proposes a six-dimensional framework for assessing and strengthening transdisciplinary knowledge validity in AI and city studies, based on an extensive analysis of the most cited research (2014--2024). Specifically, the framework classifies research orientations according to ontological, epistemological, methodological, teleological, axiological, and valorization dimensions. Our findings show a predominance of perspectives aligned with critical realism (ontological), positivism (epistemological), analytical methods (methodological), consequentialism (teleological), epistemic values (axiological), and social/economic valorization. Less common stances, such as idealism, mixed methods, and cultural valorization, are also examined for their potential to enrich knowledge production. We highlight how early career researchers and transdisciplinary teams can leverage this framework to reconcile divergent disciplinary viewpoints and promote socially accountable outcomes.

We give a conceptual treatment of the notion of joints, marginals, and independence in the setting of categorical probability. This is achieved by endowing the usual probability monads (like the Giry monad) with a monoidal and an opmonoidal structure, mutually compatible (i.e. a bimonoidal structure). If the underlying monoidal category is cartesian monoidal, a bimonoidal structure is given uniquely by a commutative strength. However, if the underlying monoidal category is not cartesian monoidal, a strength is not enough to guarantee all the desired properties of joints and marginals. A bimonoidal structure is then the correct requirement for the more general case. We explain the theory and the operational interpretation, with the help of the graphical calculus for monoidal categories. We give a definition of stochastic independence based on the bimonoidal structure, compatible with the intuition and with other approaches in the literature for cartesian monoidal categories. We then show as an example that the Kantorovich monad on the category of complete metric spaces is a bimonoidal monad for a non-cartesian monoidal structure.

This paper aims to study Ext-groups between certain functors defined on the category of finitely generated free groups. Rational Ext-groups between the abelianization functor and its symmetric powers are known, and are almost always equal to zero. Recently, using homotopical methods, Arone constructed an explicit bounded complex whose homology corresponds to the integral Ext-groups between the abelianization functor and its symmetric powers. The homology of this complex is far from being trivial. Using this complex, we explicitly calculate some of these Ext-groups. More precisely, we compute Ext^1, Ext^2, Ext^{d-1} and Ext^{d-2} between the abelianization functor and its dth symmetric power. We further explain how Arone's complex can be obtained from an explicit projective resolution of the abelianization functor. We compare our results with the computation of Ext-groups between functors from finitely generated free abelian groups, obtained by Franjou and Pirashvili. In particular, we obtain that the composition with the abelianization functor induces an isomorphism for the Ext^1 considered in this paper.

The traditional i.i.d.-based learning paradigm faces inherent challenges in addressing causal relationships, which has become increasingly evident with the rise of applications in causal representation learning. Our understanding of causality naturally requires a perspective as the creator rather than observer, as the ``what...if'' questions only hold within the possible world we conceive. The traditional perspective limits capturing dynamic causal outcomes and leads to compensatory efforts such as the reliance on hidden confounders. This paper lays the groundwork for the new perspective, which enables the relation-first modeling paradigm for causality. Also, it introduces the Relation-Indexed Representation Learning (RIRL) as a practical implementation, supported by experiments that validate its efficacy.

For any {rm E}-rigid presentation e, we construct an orthogonal projection functor to {rm rep}(e^perp) left adjoint to the natural embedding. We establish a bijection between presentations in {rm rep}(e^perp) and presentations compatible with e. For quivers with potentials, we show that {rm rep}(e^perp) forms a module category of another quiver with potential. We derive mutation formulas for the delta-vectors of positive and negative complements and the dimension vectors of simple modules in {rm rep}(e^perp), enabling an algorithm to find the projected quiver with potential. Additionally, we introduce a modified projection for quivers with potentials that preserves general presentations. For applications to cluster algebras, we establish a connection to the stabilization functors.

Every known artificial deep neural network (DNN) corresponds to an object in a canonical Grothendieck's topos; its learning dynamic corresponds to a flow of morphisms in this topos. Invariance structures in the layers (like CNNs or LSTMs) correspond to Giraud's stacks. This invariance is supposed to be responsible of the generalization property, that is extrapolation from learning data under constraints. The fibers represent pre-semantic categories (Culioli, Thom), over which artificial languages are defined, with internal logics, intuitionist, classical or linear (Girard). Semantic functioning of a network is its ability to express theories in such a language for answering questions in output about input data. Quantities and spaces of semantic information are defined by analogy with the homological interpretation of Shannon's entropy of P.Baudot and D.Bennequin in 2015). They generalize the measures found by Carnap and Bar-Hillel (1952). Amazingly, the above semantical structures are classified by geometric fibrant objects in a closed model category of Quillen, then they give rise to homotopical invariants of DNNs and of their semantic functioning. Intentional type theories (Martin-Loef) organize these objects and fibrations between them. Information contents and exchanges are analyzed by Grothendieck's derivators.

Informally, the 'linear representation hypothesis' is the idea that high-level concepts are represented linearly as directions in some representation space. In this paper, we address two closely related questions: What does "linear representation" actually mean? And, how do we make sense of geometric notions (e.g., cosine similarity or projection) in the representation space? To answer these, we use the language of counterfactuals to give two formalizations of "linear representation", one in the output (word) representation space, and one in the input (sentence) space. We then prove these connect to linear probing and model steering, respectively. To make sense of geometric notions, we use the formalization to identify a particular (non-Euclidean) inner product that respects language structure in a sense we make precise. Using this causal inner product, we show how to unify all notions of linear representation. In particular, this allows the construction of probes and steering vectors using counterfactual pairs. Experiments with LLaMA-2 demonstrate the existence of linear representations of concepts, the connection to interpretation and control, and the fundamental role of the choice of inner product.

Understanding perspective is fundamental to human visual perception, yet the extent to which multimodal large language models (MLLMs) internalize perspective geometry remains unclear. We introduce MMPerspective, the first benchmark specifically designed to systematically evaluate MLLMs' understanding of perspective through 10 carefully crafted tasks across three complementary dimensions: Perspective Perception, Reasoning, and Robustness. Our benchmark comprises 2,711 real-world and synthetic image instances with 5,083 question-answer pairs that probe key capabilities, such as vanishing point perception and counting, perspective type reasoning, line relationship understanding in 3D space, invariance to perspective-preserving transformations, etc. Through a comprehensive evaluation of 43 state-of-the-art MLLMs, we uncover significant limitations: while models demonstrate competence on surface-level perceptual tasks, they struggle with compositional reasoning and maintaining spatial consistency under perturbations. Our analysis further reveals intriguing patterns between model architecture, scale, and perspective capabilities, highlighting both robustness bottlenecks and the benefits of chain-of-thought prompting. MMPerspective establishes a valuable testbed for diagnosing and advancing spatial understanding in vision-language systems. Resources available at: https://yunlong10.github.io/MMPerspective/

Human interactions are deeply rooted in the interplay of thoughts, beliefs, and desires made possible by Theory of Mind (ToM): our cognitive ability to understand the mental states of ourselves and others. Although ToM may come naturally to us, emulating it presents a challenge to even the most advanced Large Language Models (LLMs). Recent improvements to LLMs' reasoning capabilities from simple yet effective prompting techniques such as Chain-of-Thought have seen limited applicability to ToM. In this paper, we turn to the prominent cognitive science theory "Simulation Theory" to bridge this gap. We introduce SimToM, a novel two-stage prompting framework inspired by Simulation Theory's notion of perspective-taking. To implement this idea on current ToM benchmarks, SimToM first filters context based on what the character in question knows before answering a question about their mental state. Our approach, which requires no additional training and minimal prompt-tuning, shows substantial improvement over existing methods, and our analysis reveals the importance of perspective-taking to Theory-of-Mind capabilities. Our findings suggest perspective-taking as a promising direction for future research into improving LLMs' ToM capabilities.

We consider three monads on Top, the category of topological spaces, which formalize topological aspects of probability and possibility in categorical terms. The first one is the Hoare hyperspace monad H, which assigns to every space its space of closed subsets equipped with the lower Vietoris topology. The second is the monad V of continuous valuations, also known as the extended probabilistic powerdomain. We construct both monads in a unified way in terms of double dualization. This reveals a close analogy between them, and allows us to prove that the operation of taking the support of a continuous valuation is a morphism of monads from V to H. In particular, this implies that every H-algebra (topological complete semilattice) is also a V-algebra. Third, we show that V can be restricted to a submonad of tau-smooth probability measures on Top. By composing these two morphisms of monads, we obtain that taking the support of a tau-smooth probability measure is also a morphism of monads.

Formal and distributional semantic models offer complementary benefits in modeling meaning. The categorical compositional distributional (DisCoCat) model of meaning of Coecke et al. (arXiv:1003.4394v1 [cs.CL]) combines aspected of both to provide a general framework in which meanings of words, obtained distributionally, are composed using methods from the logical setting to form sentence meaning. Concrete consequences of this general abstract setting and applications to empirical data are under active study (Grefenstette et al., arxiv:1101.0309; Grefenstette and Sadrzadeh, arXiv:1106.4058v1 [cs.CL]). . In this paper, we extend this study by examining transitive verbs, represented as matrices in a DisCoCat. We discuss three ways of constructing such matrices, and evaluate each method in a disambiguation task developed by Grefenstette and Sadrzadeh (arXiv:1106.4058v1 [cs.CL]).

Recent advances in artificial intelligence reveal the limits of purely predictive systems and call for a shift toward causal and collaborative reasoning. Drawing inspiration from the revolution of Grothendieck in mathematics, we introduce the relativity of causal knowledge, which posits structural causal models (SCMs) are inherently imperfect, subjective representations embedded within networks of relationships. By leveraging category theory, we arrange SCMs into a functor category and show that their observational and interventional probability measures naturally form convex structures. This result allows us to encode non-intervened SCMs with convex spaces of probability measures. Next, using sheaf theory, we construct the network sheaf and cosheaf of causal knowledge. These structures enable the transfer of causal knowledge across the network while incorporating interventional consistency and the perspective of the subjects, ultimately leading to the formal, mathematical definition of relative causal knowledge.

Knowing others' intentions and taking others' perspectives are two core components of human intelligence that are considered to be instantiations of theory-of-mind. Infiltrating machines with these abilities is an important step towards building human-level artificial intelligence. Here, to investigate intentionality understanding and level-2 perspective-taking in Vision Language Models (VLMs), we constructed the IntentBench and PerspectBench, which together contains over 300 cognitive experiments grounded in real-world scenarios and classic cognitive tasks. We found VLMs achieving high performance on intentionality understanding but low performance on level-2 perspective-taking. This suggests a potential dissociation between simulation-based and theory-based theory-of-mind abilities in VLMs, highlighting the concern that they are not capable of using model-based reasoning to infer others' mental states. See https://growing-ai-like-a-child.github.io/{Website}

Extracting causal relationships from observed correlations is a growing area in probabilistic reasoning, originating with the seminal work of Pearl and others from the early 1990s. This paper develops a new, categorically oriented view based on a clear distinction between syntax (string diagrams) and semantics (stochastic matrices), connected via interpretations as structure-preserving functors. A key notion in the identification of causal effects is that of an intervention, whereby a variable is forcefully set to a particular value independent of any prior propensities. We represent the effect of such an intervention as an endofunctor which performs `string diagram surgery' within the syntactic category of string diagrams. This diagram surgery in turn yields a new, interventional distribution via the interpretation functor. While in general there is no way to compute interventional distributions purely from observed data, we show that this is possible in certain special cases using a calculational tool called comb disintegration. We demonstrate the use of this technique on a well-known toy example, where we predict the causal effect of smoking on cancer in the presence of a confounding common cause. After developing this specific example, we show this technique provides simple sufficient conditions for computing interventions which apply to a wide variety of situations considered in the causal inference literature.

The reverse derivative is a fundamental operation in machine learning and automatic differentiation. This paper gives a direct axiomatization of a category with a reverse derivative operation, in a similar style to that given by Cartesian differential categories for a forward derivative. Intriguingly, a category with a reverse derivative also has a forward derivative, but the converse is not true. In fact, we show explicitly what a forward derivative is missing: a reverse derivative is equivalent to a forward derivative with a dagger structure on its subcategory of linear maps. Furthermore, we show that these linear maps form an additively enriched category with dagger biproducts.

The common toxicity and societal bias in contents generated by large language models (LLMs) necessitate strategies to reduce harm. Present solutions often demand white-box access to the model or substantial training, which is impractical for cutting-edge commercial LLMs. Moreover, prevailing prompting methods depend on external tool feedback and fail to simultaneously lessen toxicity and bias. Motivated by social psychology principles, we propose a novel strategy named perspective-taking prompting (\textsc{PeT)} that inspires LLMs to integrate diverse human perspectives and self-regulate their responses. This self-correction mechanism can significantly diminish toxicity (up to 89%) and bias (up to 73%) in LLMs' responses. Rigorous evaluations and ablation studies are conducted on two commercial LLMs (ChatGPT and GLM) and three open-source LLMs, revealing PeT's superiority in producing less harmful responses, outperforming five strong baselines.

With infinitely many high-quality data points, infinite computational power, an infinitely large foundation model with a perfect training algorithm and guaranteed zero generalization error on the pretext task, can the model be used for everything? This question cannot be answered by the existing theory of representation, optimization or generalization, because the issues they mainly investigate are assumed to be nonexistent here. In this paper, we show that category theory provides powerful machinery to answer this question. We have proved three results. The first one limits the power of prompt-based learning, saying that the model can solve a downstream task with prompts if and only if the task is representable. The second one says fine tuning does not have this limit, as a foundation model with the minimum required power (up to symmetry) can theoretically solve downstream tasks for the category defined by pretext task, with fine tuning and enough resources. Our final result can be seen as a new type of generalization theorem, showing that the foundation model can generate unseen objects from the target category (e.g., images) using the structural information from the source category (e.g., texts). Along the way, we provide a categorical framework for supervised and self-supervised learning, which might be of independent interest.

We consider the problem of characterizing derived endomorphism algebras of simple objects in length categories up to quasi-isomorphism. We give such a characterization for module categories, abelian categories, exact categories, as well as, for certain differential graded analogues of them. It turns out that the property of being d-complicial (dgeq 1), in the sense of Lurie, of the involved simple-minded collections plays a central role. We also explain how this characterization can be interpreted as a coherent generation property for any minimal A_{infty}-model of the derived endomorphism algebra. Along the way, we propose a notion of length exact differential graded categories and explain how they relate to length abelian d-truncated differential graded categories, generalizing results of Enomoto.

In this work we take a Category Theoretic perspective on the relationship between probabilistic modeling and function approximation. We begin by defining two extensions of function composition to stochastic process subordination: one based on the co-Kleisli category under the comonad (Omega x -) and one based on the parameterization of a category with a Lawvere theory. We show how these extensions relate to the category Stoch and other Markov Categories. Next, we apply the Para construction to extend stochastic processes to parameterized statistical models and we define a way to compose the likelihood functions of these models. We conclude with a demonstration of how the Maximum Likelihood Estimation procedure defines an identity-on-objects functor from the category of statistical models to the category of Learners. Code to accompany this paper can be found at https://github.com/dshieble/Categorical_Stochastic_Processes_and_Likelihood

Low-dimensional projections of text embeddings support visual analysis of document collections, but their spatial organization may not reflect the relationships an analyst intends to examine. Existing semantic interaction approaches encode semantic intent indirectly through geometric constraints or model updates, limiting interpretability and flexibility. We introduce LLM-augmented semantic steering, which enables analysts to express semantic intent by grouping a small set of example documents within the projection. A large language model externalizes this intent as natural-language representations and selectively extends it to related documents; the resulting semantic information is then incorporated into document representations via text augmentation or embedding-level blending, without retraining the underlying models. A case study illustrates how the same corpus can be reorganized from different semantic perspectives, while simulation-based evaluation shows that semantic steering improves global and local alignment with target semantic structures using only minimal interaction. Embedding-level blending further enables continuous and controllable steering of projection layouts. These results position projection spaces as intent-dependent semantic workspaces that can be reshaped through explicit, interpretable, language-mediated interaction.

The ability of language models to learn a task from a few examples in context has generated substantial interest. Here, we provide a perspective that situates this type of supervised few-shot learning within a much broader spectrum of meta-learned in-context learning. Indeed, we suggest that any distribution of sequences in which context non-trivially decreases loss on subsequent predictions can be interpreted as eliciting a kind of in-context learning. We suggest that this perspective helps to unify the broad set of in-context abilities that language models exhibit x2014 such as adapting to tasks from instructions or role play, or extrapolating time series. This perspective also sheds light on potential roots of in-context learning in lower-level processing of linguistic dependencies (e.g. coreference or parallel structures). Finally, taking this perspective highlights the importance of generalization, which we suggest can be studied along several dimensions: not only the ability to learn something novel, but also flexibility in learning from different presentations, and in applying what is learned. We discuss broader connections to past literature in meta-learning and goal-conditioned agents, and other perspectives on learning and adaptation. We close by suggesting that research on in-context learning should consider this broader spectrum of in-context capabilities and types of generalization.

Language builds discourse contexts other than the actual: a painting, a belief, a memory, a hypothetical. Each is a mental space in which the same entity can take a different value, as when a flower is red in reality but purple in a portrait. Formal semantics keeps these contexts apart because their logics differ (modal, temporal, doxastic, depictive); Fauconnier's mental-space theory treats them as one space-building operation. We ask which of these a transformer language model implements, and find a mechanistic version of Fauconnier's unification. The model uses one router/slot format across the inventory: a reusable value slot stores attributed content, and a causally manipulable router (the space index) selects which space is read. A subspace trained with Distributed Alignment Search to control one space type, counterfactual, belief, fictional, or temporal, also controls the others, well above a random floor, on three model families; belief, which formal semantics marks as a distinct case, is not specially separated. The router is low-rank, composes additively with entity identity, and acts through a few late-layer heads. Two further results show the mechanism drives inference and composes: a subspace trained on a rule-derived conclusion flips what the model infers while dissociating from what it reports, and composing space-builders mints a fresh router over the shared slot. This paper establishes the cross-type generality. A companion paper develops belief in depth, because of its special status in philosophy, psychology, and linguistics (epistemology, theory of mind, and propositional attitude reports).

Driven by vast and diverse textual data, large language models (LLMs) have demonstrated impressive performance across numerous natural language processing (NLP) tasks. Yet, a critical question persists: does their generalization arise from mere memorization of training data or from deep semantic understanding? To investigate this, we propose a bi-perspective evaluation framework to assess LLMs' scenario cognition - the ability to link semantic scenario elements with their arguments in context. Specifically, we introduce a novel scenario-based dataset comprising diverse textual descriptions of fictional facts, annotated with scenario elements. LLMs are evaluated through their capacity to answer scenario-related questions (model output perspective) and via probing their internal representations for encoded scenario elements-argument associations (internal representation perspective). Our experiments reveal that current LLMs predominantly rely on superficial memorization, failing to achieve robust semantic scenario cognition, even in simple cases. These findings expose critical limitations in LLMs' semantic understanding and offer cognitive insights for advancing their capabilities.

Category theory has been successfully applied in various domains of science, shedding light on universal principles unifying diverse phenomena and thereby enabling knowledge transfer between them. Applications to machine learning have been pursued recently, and yet there is still a gap between abstract mathematical foundations and concrete applications to machine learning tasks. In this paper we introduce DisCoPyro as a categorical structure learning framework, which combines categorical structures (such as symmetric monoidal categories and operads) with amortized variational inference, and can be applied, e.g., in program learning for variational autoencoders. We provide both mathematical foundations and concrete applications together with comparison of experimental performance with other models (e.g., neuro-symbolic models). We speculate that DisCoPyro could ultimately contribute to the development of artificial general intelligence.

Coecke, Sadrzadeh, and Clark (arXiv:1003.4394v1 [cs.CL]) developed a compositional model of meaning for distributional semantics, in which each word in a sentence has a meaning vector and the distributional meaning of the sentence is a function of the tensor products of the word vectors. Abstractly speaking, this function is the morphism corresponding to the grammatical structure of the sentence in the category of finite dimensional vector spaces. In this paper, we provide a concrete method for implementing this linear meaning map, by constructing a corpus-based vector space for the type of sentence. Our construction method is based on structured vector spaces whereby meaning vectors of all sentences, regardless of their grammatical structure, live in the same vector space. Our proposed sentence space is the tensor product of two noun spaces, in which the basis vectors are pairs of words each augmented with a grammatical role. This enables us to compare meanings of sentences by simply taking the inner product of their vectors.

Over the past two decades machine learning has permeated almost every realm of technology. At the same time, many researchers have begun using category theory as a unifying language, facilitating communication between different scientific disciplines. It is therefore unsurprising that there is a burgeoning interest in applying category theory to machine learning. We aim to document the motivations, goals and common themes across these applications. We touch on gradient-based learning, probability, and equivariant learning.

Drawing on contemporary pragmatist philosophy and linguistic theories on cognition, meaning, and communication, this paper presents a dynamic, metasemantic-metapragmatic taxonomy for grounding and conceptualizing human-like multimodal communicative alignment. The framework is rooted in contemporary developments of the three basic communicative capacities initially identified by American logician and pragmatist philosopher Charles Sanders Peirce: iconic (sensory and perceptual qualities), indexical (contextual and sociocultural associations), and rule-like (symbolic and intuitive reasoning). Expanding on these developments, I introduce the concept of indexical contextualization and propose the principle of "contextualization directionality" for characterizing the crucial metapragmatic capacity for maintaining, navigating, or transitioning between semantic and pragmatic modes of multimodal communication. I contend that current cognitive-social computational and engineering methodologies disproportionately emphasize the semantic/metasemantic domain, overlooking the pivotal role of metapragmatic indexicality in traversing the semantic-pragmatic spectrum of communication. The framework's broader implications for intentionality, identity, affect, and ethics in within-modal and cross-modal human-machine alignment are also discussed.

We present a framework for perspective-aware reasoning in vision-language models (VLMs) through mental imagery simulation. Perspective-taking, the ability to perceive an environment or situation from an alternative viewpoint, is a key benchmark for human-level visual understanding, essential for environmental interaction and collaboration with autonomous agents. Despite advancements in spatial reasoning within VLMs, recent research has shown that modern VLMs significantly lack perspective-aware reasoning capabilities and exhibit a strong bias toward egocentric interpretations. To bridge the gap between VLMs and human perception, we focus on the role of mental imagery, where humans perceive the world through abstracted representations that facilitate perspective shifts. Motivated by this, we propose a framework for perspective-aware reasoning, named Abstract Perspective Change (APC), that effectively leverages vision foundation models, such as object detection, segmentation, and orientation estimation, to construct scene abstractions and enable perspective transformations. Our experiments on synthetic and real-image benchmarks, compared with various VLMs, demonstrate significant improvements in perspective-aware reasoning with our framework, further outperforming fine-tuned spatial reasoning models and novel-view-synthesis-based approaches.

While large language models (LLMs) appear to be increasingly capable of solving compositional tasks, it is an open question whether they do so using compositional mechanisms. In this work, we investigate how feedforward LLMs solve two-hop factual recall tasks, which can be expressed compositionally as g(f(x)). We first confirm that modern LLMs continue to suffer from the "compositionality gap": i.e. their ability to compute both z = f(x) and y = g(z) does not entail their ability to compute the composition y = g(f(x)). Then, using logit lens on their residual stream activations, we identify two processing mechanisms, one which solves tasks compositionally, computing f(x) along the way to computing g(f(x)), and one which solves them directly, without any detectable signature of the intermediate variable f(x). Finally, we find that which mechanism is employed appears to be related to the embedding space geometry, with the idiomatic mechanism being dominant in cases where there exists a linear mapping from x to g(f(x)) in the embedding spaces. We fully release our data and code at: https://github.com/apoorvkh/composing-functions .

Scientific studies of consciousness rely on objects whose existence is assumed to be independent of any consciousness. On the contrary, we assume consciousness to be fundamental, and that one of the main features of consciousness is characterized as being other-dependent. We set up a framework which naturally subsumes this feature by defining a compact closed category where morphisms represent conscious processes. These morphisms are a composition of a set of generators, each being specified by their relations with other generators, and therefore co-dependent. The framework is general enough and fits well into a compositional model of consciousness. Interestingly, we also show how our proposal may become a step towards avoiding the hard problem of consciousness, and thereby address the combination problem of conscious experiences.

We construct S-linear semiorthogonal decompositions of derived categories of smooth Fano threefold fibrations X/S with relative Picard rank 1 and rational geometric fibers and discuss how the structure of components of these decompositions is related to rationality properties of X/S.

We provide a construction for categorical representation learning and introduce the foundations of "categorifier". The central theme in representation learning is the idea of everything to vector. Every object in a dataset S can be represented as a vector in R^n by an encoding map E: Obj(S)toR^n. More importantly, every morphism can be represented as a matrix E: Hom(S)toR^{n}_{n}. The encoding map E is generally modeled by a deep neural network. The goal of representation learning is to design appropriate tasks on the dataset to train the encoding map (assuming that an encoding is optimal if it universally optimizes the performance on various tasks). However, the latter is still a set-theoretic approach. The goal of the current article is to promote the representation learning to a new level via a category-theoretic approach. As a proof of concept, we provide an example of a text translator equipped with our technology, showing that our categorical learning model outperforms the current deep learning models by 17 times. The content of the current article is part of the recent US patent proposal (patent application number: 63110906).

We present an algorithm that constructs Calabi-Yau threefold orientifolds and their F-theory uplifts to elliptically-fibered Calabi-Yau fourfolds, embedded in toric varieties at codimension one and two respectively. The resulting Calabi-Yau fourfolds arise from triangulations of 6d reflexive polytopes -- which our method constructs from orientifold data -- and are smooth away from isolated terminal singularities. For many of our fourfolds, the construction of the mirror manifold is immediate, enabling the computation of fourfold periods, and thus the seven-brane superpotential. We present multiple examples that demonstrate these capabilities. Our algorithms work with CYTools and are available through a GitHub repository.

Backward Filtering Forward Guiding (BFFG) is a bidirectional algorithm proposed in Mider et al. [2021] and studied more in depth in a general setting in Van der Meulen and Schauer [2022]. In category theory, optics have been proposed for modelling systems with bidirectional data flow. We connect BFFG with optics and prove that different ways of composing the building blocks of BFFG correspond to equivalent optics.

DisCoPy is a Python toolkit for computing with monoidal categories. It comes with two flexible data structures for string diagrams: the first one for planar monoidal categories based on lists of layers, the second one for symmetric monoidal categories based on cospans of hypergraphs. Algorithms for functor application then allow to translate string diagrams into code for numerical computation, be it differentiable, probabilistic or quantum. This report gives an overview of the library and the new developments released in its version 1.0. In particular, we showcase the implementation of diagram equality for a large fragment of the hierarchy of graphical languages for monoidal categories, as well as a new syntax for defining string diagrams as Python functions.

This paper is a very non-rigorous, loose, and extremely basic introduction to sheaves. This is meant to be a a guide to gaining intuition about sheaves, what they look like, and how they work, so that after reading this paper, someone can jump into the extremely abstract definitions and examples seen in textbooks with at least some idea of what is going on. Most of this material is inspired and built from the work of Dr. Michael Robinson, and that of Dr. Robert Ghrist and Dr. Jakob Hansen, as well as Dr. Justin Curry's PhD thesis, who are some of the only applied sheaf theorists out there and they do an amazing job of explaining sheaves in a concrete way through their research. The rest of this paper is populated by mathematical definitions found in textbooks that I have stretched from two lines into multiple pages, as well as some analogies for thinking of sheaves I have thought of myself. This paper only assumes knowledge of basic linear algebra, basic group theory, and the very fundamentals of topology. If there is anything in the setup that you do not understand it is probably a quick Wikipedia search away. I hope this paper provides insight, intuition, and helpful examples of why sheaves are such powerful tools in both math and science.

In order to avoid well-know paradoxes associated with self-referential definitions, higher-order dependent type theories stratify the theory using a countably infinite hierarchy of universes (also known as sorts), Type_0 : Type_1 : cdots . Such type systems are called cumulative if for any type A we have that A : Type_{i} implies A : Type_{i+1}. The predicative calculus of inductive constructions (pCIC) which forms the basis of the Coq proof assistant, is one such system. In this paper we present and establish the soundness of the predicative calculus of cumulative inductive constructions (pCuIC) which extends the cumulativity relation to inductive types.

We introduce Drivelology, a unique linguistic phenomenon characterised as "nonsense with depth", utterances that are syntactically coherent yet pragmatically paradoxical, emotionally loaded, or rhetorically subversive. While such expressions may resemble surface-level nonsense, they encode implicit meaning requiring contextual inference, moral reasoning, or emotional interpretation. We find that current large language models (LLMs), despite excelling at many natural language processing (NLP) tasks, consistently fail to grasp the layered semantics of Drivelological text. To investigate this, we construct a small but diverse benchmark dataset of over 1,200 meticulously curated examples, with select instances in English, Mandarin, Spanish, French, Japanese, and Korean. Annotation was especially challenging: each of the examples required careful expert review to verify that it truly reflected Drivelological characteristics. The process involved multiple rounds of discussion and adjudication to address disagreements, highlighting the subtle and subjective nature of the Drivelology. We evaluate a range of LLMs on classification, generation, and reasoning tasks. Our results reveal clear limitations of LLMs: models often confuse Drivelology with shallow nonsense, produce incoherent justifications, or miss the implied rhetorical function altogether. These findings highlight a deeper representational gap in LLMs' pragmatic understanding and challenge the assumption that statistical fluency implies cognitive comprehension. We release our dataset and code to facilitate further research in modelling linguistic depth beyond surface-level coherence.

In Marius von Senden's Space and Sight, a newly sighted blind patient describes the experience of a corner as lemon-like, because corners "prick" sight like lemons prick the tongue. Prickliness, here, is a dimension in the feature space of sensory experience, an effect of the perceived on the perceiver that arises where the two interact. In the account of the newly sighted, an effect familiar from one interaction translates to a novel context. Perception serves as the vehicle for generalization, in that an effect shared across different experiences produces a concrete abstraction grounded in those experiences. Cezanne and the post-impressionists, fluent in the language of experience translation, realized that the way to paint a concrete form that best reflected reality was to paint not what they saw, but what it was like to see. We envision a future of creation using AI where what it is like to see is replicable, transferrable, manipulable - part of the artist's palette that is both grounded in a particular context, and generalizable beyond it. An active line of research maps human-interpretable features onto directions in GAN latent space. Supervised and self-supervised approaches that search for anticipated directions or use off-the-shelf classifiers to drive image manipulation in embedding space are limited in the variety of features they can uncover. Unsupervised approaches that discover useful new directions show that the space of perceptually meaningful directions is nowhere close to being fully mapped. As this space is broad and full of creative potential, we want tools for direction discovery that capture the richness and generalizability of human perception. Our approach puts creators in the discovery loop during real-time tool use, in order to identify directions that are perceptually meaningful to them, and generate interpretable image translations along those directions.

Small sample sizes are common in many disciplines, which necessitates pooling roughly similar datasets across multiple institutions to study weak but relevant associations between images and disease outcomes. Such data often manifest shift/imbalance in covariates (i.e., secondary non-imaging data). Controlling for such nuisance variables is common within standard statistical analysis, but the ideas do not directly apply to overparameterized models. Consequently, recent work has shown how strategies from invariant representation learning provides a meaningful starting point, but the current repertoire of methods is limited to accounting for shifts/imbalances in just a couple of covariates at a time. In this paper, we show how viewing this problem from the perspective of Category theory provides a simple and effective solution that completely avoids elaborate multi-stage training pipelines that would otherwise be needed. We show the effectiveness of this approach via extensive experiments on real datasets. Further, we discuss how this style of formulation offers a unified perspective on at least 5+ distinct problem settings, from self-supervised learning to matching problems in 3D reconstruction.

This note is about an extension of the Kodaira-Spencer functional to Calabi-Yau manifolds of any dimension.

We propose a new definition of higher-order DisCoCat (categorical compositional distributional) models where the meaning of a word is not a diagram, but a diagram-valued higher-order function. Our models can be seen as a variant of Montague semantics based on a lambda calculus where the primitives act on string diagrams rather than logical formulae. As a special case, we show how to translate from the Lambek calculus into Peirce's system beta for first-order logic. This allows us to give a purely diagrammatic treatment of higher-order and non-linear processes in natural language semantics: adverbs, prepositions, negation and quantifiers. The theoretical definition presented in this article comes with a proof-of-concept implementation in DisCoPy, the Python library for string diagrams.

Large language models (LLMs) have been reported to linearly encode truthfulness, yet recent work questions this finding's generality. We reconcile these views with the truthfulness spectrum hypothesis: the representational space contains directions ranging from broadly domain-general to narrowly domain-specific. To test this hypothesis, we systematically evaluate probe generalization across five truth types (definitional, empirical, logical, fictional, and ethical), sycophantic and expectation-inverted lying, and existing honesty benchmarks. Linear probes generalize well across most domains but fail on sycophantic and expectation-inverted lying. Yet training on all domains jointly recovers strong performance, confirming that domain-general directions exist despite poor pairwise transfer. The geometry of probe directions explains these patterns: Mahalanobis cosine similarity between probes near-perfectly predicts cross-domain generalization (R^2=0.98). Concept-erasure methods further isolate truth directions that are (1) domain-general, (2) domain-specific, or (3) shared only across particular domain subsets. Causal interventions reveal that domain-specific directions steer more effectively than domain-general ones. Finally, post-training reshapes truth geometry, pushing sycophantic lying further from other truth types, suggesting a representational basis for chat models' sycophantic tendencies. Together, our results support the truthfulness spectrum hypothesis: truth directions of varying generality coexist in representational space, with post-training reshaping their geometry. Code for all experiments is provided in https://github.com/zfying/truth_spec.

We present a study on how and where personas -- defined by distinct sets of human characteristics, values, and beliefs -- are encoded in the representation space of large language models (LLMs). Using a range of dimension reduction and pattern recognition methods, we first identify the model layers that show the greatest divergence in encoding these representations. We then analyze the activations within a selected layer to examine how specific personas are encoded relative to others, including their shared and distinct embedding spaces. We find that, across multiple pre-trained decoder-only LLMs, the analyzed personas show large differences in representation space only within the final third of the decoder layers. We observe overlapping activations for specific ethical perspectives -- such as moral nihilism and utilitarianism -- suggesting a degree of polysemy. In contrast, political ideologies like conservatism and liberalism appear to be represented in more distinct regions. These findings help to improve our understanding of how LLMs internally represent information and can inform future efforts in refining the modulation of specific human traits in LLM outputs. Warning: This paper includes potentially offensive sample statements.

This work explores the hypothesis that subjectively attributed meaning constitutes the phenomenal content of conscious experience. That is, phenomenal content is semantic. This form of subjective meaning manifests as an intrinsic and non-representational character of qualia. Empirically, subjective meaning is ubiquitous in conscious experiences. We point to phenomenological studies that lend evidence to support this. Furthermore, this notion of meaning closely relates to what Frege refers to as "sense", in metaphysics and philosophy of language. It also aligns with Peirce's "interpretant", in semiotics. We discuss how Frege's sense can also be extended to the raw feels of consciousness. Sense and reference both play a role in phenomenal experience. Moreover, within the context of the mind-matter relation, we provide a formalization of subjective meaning associated to one's mental representations. Identifying the precise maps between the physical and mental domains, we argue that syntactic and semantic structures transcend language, and are realized within each of these domains. Formally, meaning is a relational attribute, realized via a map that interprets syntactic structures of a formal system within an appropriate semantic space. The image of this map within the mental domain is what is relevant for experience, and thus comprises the phenomenal content of qualia. We conclude with possible implications this may have for experience-based theories of consciousness.

Bayes' rule tells us how to invert a causal process in order to update our beliefs in light of new evidence. If the process is believed to have a complex compositional structure, we may ask whether composing the inversions of the component processes gives the same belief update as the inversion of the whole. We answer this question affirmatively, showing that the relevant compositional structure is precisely that of the lens pattern, and that we can think of Bayesian inversion as a particular instance of a state-dependent morphism in a corresponding fibred category. We define a general notion of (mixed) Bayesian lens, and discuss the (un)lawfulness of these lenses when their contravariant components are exact Bayesian inversions. We prove our main result both abstractly and concretely, for both discrete and continuous states, taking care to illustrate the common structures.

We present experimental results to explore a form of bivariate glyphs for representing large-magnitude-range vectors. The glyphs meet two conditions: (1) two visual dimensions are separable; and (2) one of the two visual dimensions uses a categorical representation (e.g., a categorical colormap). We evaluate how much these two conditions determine the bivariate glyphs' effectiveness. The first experiment asks participants to perform three local tasks requiring reading no more than two glyphs. The second experiment scales up the search space in global tasks when participants must look at the entire scene of hundreds of vector glyphs to get an answer. Our results support that the first condition is necessary for local tasks when a few items are compared. But it is not enough to understand a large amount of data. The second condition is necessary for perceiving global structures of examining very complex datasets. Participants' comments reveal that the categorical features in the bivariate glyphs trigger emergent optimal viewers' behaviors. This work contributes to perceptually accurate glyph representations for revealing patterns from large scientific results. We release source code, quantum physics data, training documents, participants' answers, and statistical analyses for reproducible science https://osf.io/4xcf5/?view_only=94123139df9c4ac984a1e0df811cd580.

In this paper, we introduce fundamental notions of homotopy theory, including homotopy excision and the Freudenthal suspension theorem. We then explore framed cobordism and its connection to stable homotopy groups of spheres through the Pontryagin-Thom construction. Using this framework, we compute the stable stems in dimensions 0, 1, and 2. This work is primarily expository, revisiting proofs from Pont1 with slight modifications incorporating modern notation. Furthermore, in the final section, we discuss 2-dimensional framed manifolds with Arf invariant one and examine why the result of Pont2 regarding π_2^S is incorrect.

The task of perspective-aware classification introduces a bottleneck in terms of parametric efficiency that did not get enough recognition in existing studies. In this article, we aim to address this issue by applying an existing architecture, the hypernetwork+adapters combination, to perspectivist classification. Ultimately, we arrive at a solution that can compete with specialized models in adopting user perspectives on hate speech and toxicity detection, while also making use of considerably fewer parameters. Our solution is architecture-agnostic and can be applied to a wide range of base models out of the box.

Commonsense causality reasoning (CCR) aims at identifying plausible causes and effects in natural language descriptions that are deemed reasonable by an average person. Although being of great academic and practical interest, this problem is still shadowed by the lack of a well-posed theoretical framework; existing work usually relies on deep language models wholeheartedly, and is potentially susceptible to confounding co-occurrences. Motivated by classical causal principles, we articulate the central question of CCR and draw parallels between human subjects in observational studies and natural languages to adopt CCR to the potential-outcomes framework, which is the first such attempt for commonsense tasks. We propose a novel framework, ROCK, to Reason O(A)bout Commonsense K(C)ausality, which utilizes temporal signals as incidental supervision, and balances confounding effects using temporal propensities that are analogous to propensity scores. The ROCK implementation is modular and zero-shot, and demonstrates good CCR capabilities.

Modern large language models (LLMs) employ various forms of logical inference, both implicitly and explicitly, when addressing reasoning tasks. Understanding how to optimally leverage these inference paradigms is critical for advancing LLMs' reasoning capabilities. This paper adopts an exploratory approach by introducing a controlled evaluation environment for analogical reasoning -- a fundamental cognitive task -- that is systematically parameterized across three dimensions: modality (textual, visual, symbolic), difficulty (easy, medium, hard), and task format (multiple-choice or free-text generation). We analyze the comparative dynamics of inductive, abductive, and deductive inference pipelines across these dimensions, and demonstrate that our findings generalize to broader in-context learning tasks. Additionally, we investigate advanced paradigms such as hypothesis selection, verification, and refinement, revealing their potential to scale up logical inference in LLM reasoning. This exploratory study provides a foundation for future research in enhancing LLM reasoning through systematic logical inference strategies.

"Sure, I am happy to generate a story for you: Captain Lyra stood at the helm of her trusty ship, the Maelstrom's Fury, gazing out at the endless sea. [...] Lyra's eyes welled up with tears as she realized the bitter truth - she had sacrificed everything for fleeting riches, and lost the love of her crew, her family, and herself." Although this story, generated by a large language model, is captivating, one may wonder -- how would the story have unfolded if the model had chosen "Captain Maeve" as the protagonist instead? We cannot know. State-of-the-art large language models are stateless -- they maintain no internal memory or state. Given a prompt, they generate a sequence of tokens as an output using an autoregressive process. As a consequence, they cannot reason about counterfactual alternatives to tokens they have generated in the past. In this work, our goal is to enhance them with this functionality. To this end, we develop a causal model of token generation that builds upon the Gumbel-Max structural causal model. Our model allows any large language model to perform counterfactual token generation at almost no cost in comparison with vanilla token generation, it is embarrassingly simple to implement, and it does not require any fine-tuning nor prompt engineering. We implement our model on Llama 3 8B-Instruct and Ministral-8B-Instruct and conduct a qualitative and a quantitative analysis of counterfactually generated text. We conclude with a demonstrative application of counterfactual token generation for bias detection, unveiling interesting insights about the model of the world constructed by large language models.

This paper explores the use of large language models (LLMs) as research tools in the history, philosophy, and sociology of science (HPSS). LLMs are remarkably effective at processing unstructured text and inferring meaning from context, offering new affordances that challenge long-standing divides between computational and interpretive methods. This raises both opportunities and challenges for HPSS, which emphasizes interpretive methodologies and understands meaning as context-dependent, ambiguous, and historically situated. We argue that HPSS is uniquely positioned not only to benefit from LLMs' capabilities but also to interrogate their epistemic assumptions and infrastructural implications. To this end, we first offer a concise primer on LLM architectures and training paradigms tailored to non-technical readers. We frame LLMs not as neutral tools but as epistemic infrastructures that encode assumptions about meaning, context, and similarity, conditioned by their training data, architecture, and patterns of use. We then examine how computational techniques enhanced by LLMs, such as structuring data, detecting patterns, and modeling dynamic processes, can be applied to support interpretive research in HPSS. Our analysis compares full-context and generative models, outlines strategies for domain and task adaptation (e.g., continued pretraining, fine-tuning, and retrieval-augmented generation), and evaluates their respective strengths and limitations for interpretive inquiry in HPSS. We conclude with four lessons for integrating LLMs into HPSS: (1) model selection involves interpretive trade-offs; (2) LLM literacy is foundational; (3) HPSS must define its own benchmarks and corpora; and (4) LLMs should enhance, not replace, interpretive methods.

Many learning algorithms have invariances: when their training data is transformed in certain ways, the function they learn transforms in a predictable manner. Here we formalize this notion using concepts from the mathematical field of category theory. The invariances that a supervised learning algorithm possesses are formalized by categories of predictor and target spaces, whose morphisms represent the algorithm's invariances, and an index category whose morphisms represent permutations of the training examples. An invariant learning algorithm is a natural transformation between two functors from the product of these categories to the category of sets, representing training datasets and learned functions respectively. We illustrate the framework by characterizing and contrasting the invariances of linear regression and ridge regression.

Psychological research consistently finds that human ratings of words across diverse semantic scales can be reduced to a low-dimensional form with relatively little information loss. We find that the semantic associations encoded in the embedding matrices of large language models (LLMs) exhibit a similar structure. We show that the projections of words on semantic directions defined by antonym pairs (e.g. kind - cruel) correlate highly with human ratings, and further find that these projections effectively reduce to a 3-dimensional subspace within LLM embeddings, closely resembling the patterns derived from human survey responses. Moreover, we find that shifting tokens along one semantic direction causes off-target effects on geometrically aligned features proportional to their cosine similarity. These findings suggest that semantic features are entangled within LLMs similarly to how they are interconnected in human language, and a great deal of semantic information, despite its apparent complexity, is surprisingly low-dimensional. Furthermore, accounting for this semantic structure may prove essential for avoiding unintended consequences when steering features.

We present our position on the elusive quest for a general-purpose framework for specifying and studying deep learning architectures. Our opinion is that the key attempts made so far lack a coherent bridge between specifying constraints which models must satisfy and specifying their implementations. Focusing on building a such a bridge, we propose to apply category theory -- precisely, the universal algebra of monads valued in a 2-category of parametric maps -- as a single theory elegantly subsuming both of these flavours of neural network design. To defend our position, we show how this theory recovers constraints induced by geometric deep learning, as well as implementations of many architectures drawn from the diverse landscape of neural networks, such as RNNs. We also illustrate how the theory naturally encodes many standard constructs in computer science and automata theory.

Modelling compositionality has been a longstanding area of research in the field of vector space semantics. The categorical approach to compositionality maps grammar onto vector spaces in a principled way, but comes under fire for requiring the formation of very high-dimensional matrices and tensors, and therefore being computationally infeasible. In this paper I show how a linear simplification of recursive neural tensor network models can be mapped directly onto the categorical approach, giving a way of computing the required matrices and tensors. This mapping suggests a number of lines of research for both categorical compositional vector space models of meaning and for recursive neural network models of compositionality.

Let V be a highest weight module over a Kac-Moody algebra g, and let conv V denote the convex hull of its weights. We determine the combinatorial isomorphism type of conv V, i.e. we completely classify the faces and their inclusions. In the special case where g is semisimple, this brings closure to a question studied by Cellini-Marietti [IMRN 2015] for the adjoint representation, and by Khare [J. Algebra 2016; Trans. Amer. Math. Soc. 2017] for most modules. The determination of faces of finite-dimensional modules up to the Weyl group action and some of their inclusions also appears in previous work of Satake [Ann. of Math. 1960], Borel-Tits [IHES Publ. Math. 1965], Vinberg [Izv. Akad. Nauk 1990], and Casselman [Austral. Math. Soc. 1997]. For any subset of the simple roots, we introduce a remarkable convex cone which we call the universal Weyl polyhedron, which controls the convex hulls of all modules parabolically induced from the corresponding Levi factor. Namely, the combinatorial isomorphism type of the cone stores the classification of faces for all such highest weight modules, as well as how faces degenerate as the highest weight gets increasingly singular. To our knowledge, this cone is new in finite and infinite type. We further answer a question of Michel Brion, by showing that the localization of conv V along a face is always the convex hull of the weights of a parabolically induced module. Finally, as we determine the inclusion relations between faces representation-theoretically from the set of weights, without recourse to convexity, we answer a similar question for highest weight modules over symmetrizable quantum groups.

While researchers are finding concepts represented as linear directions in language models, a bag of linear directions fails to capture relational structure. To better understand this dichotomy, we study a model with known linear representations, but trained in a highly structured domain -- the board game Othello. While the model's internal board-state representation is linearly decodable, we find additional structure in the form of tensor product representations (TPRs). We train TPR probes to recover shared structure amongst the linear probes, yielding a factorization into square-embeddings, color-embeddings, and a binding matrix that composes them to construct the model's board-state representation. We find geometric signatures within the weights of our TPR probe that align with the structure of the board, but perhaps more importantly, that the linear probes can be recovered directly from the parameters of our TPR probe. Our findings suggest that directional representations may be projections of more structured underlying representations.

Visual perspective-taking (VPT), the ability to understand the viewpoint of another person, enables individuals to anticipate the actions of other people. For instance, a driver can avoid accidents by assessing what pedestrians see. Humans typically develop this skill in early childhood, but it remains unclear whether the recently emerging Vision Language Models (VLMs) possess such capability. Furthermore, as these models are increasingly deployed in the real world, understanding how they perform nuanced tasks like VPT becomes essential. In this paper, we introduce two manually curated datasets, Isle-Bricks and Isle-Dots for testing VPT skills, and we use it to evaluate 12 commonly used VLMs. Across all models, we observe a significant performance drop when perspective-taking is required. Additionally, we find performance in object detection tasks is poorly correlated with performance on VPT tasks, suggesting that the existing benchmarks might not be sufficient to understand this problem. The code and the dataset will be available at https://sites.google.com/view/perspective-taking

This paper reformulates a classical result in probability theory from the 1930s in modern categorical terms: de Finetti's representation theorem is redescribed as limit statement for a chain of finite spaces in the Kleisli category of the Giry monad. This new limit is used to identify among exchangeable coalgebras the final one.

Understanding how semantic meaning is encoded in the representation spaces of large language models is a fundamental problem in interpretability. In this paper, we study the two foundational questions in this area. First, how are categorical concepts, such as {'mammal', 'bird', 'reptile', 'fish'}, represented? Second, how are hierarchical relations between concepts encoded? For example, how is the fact that 'dog' is a kind of 'mammal' encoded? We show how to extend the linear representation hypothesis to answer these questions. We find a remarkably simple structure: simple categorical concepts are represented as simplices, hierarchically related concepts are orthogonal in a sense we make precise, and (in consequence) complex concepts are represented as polytopes constructed from direct sums of simplices, reflecting the hierarchical structure. We validate these theoretical results on the Gemma large language model, estimating representations for 957 hierarchically related concepts using data from WordNet.

Hyperbolic space has proven to be well-suited for capturing hierarchical relations in data, such as trees and directed acyclic graphs. Prior work introduced the concept of entailment cones, which uses partial orders defined by nested cones in the Poincar\'e ball to model hierarchies. Here, we introduce the ``shadow cones" framework, a physics-inspired entailment cone construction. Specifically, we model partial orders as subset relations between shadows formed by a light source and opaque objects in hyperbolic space. The shadow cones framework generalizes entailment cones to a broad class of formulations and hyperbolic space models beyond the Poincar\'e ball. This results in clear advantages over existing constructions: for example, shadow cones possess better optimization properties over constructions limited to the Poincar\'e ball. Our experiments on datasets of various sizes and hierarchical structures show that shadow cones consistently and significantly outperform existing entailment cone constructions. These results indicate that shadow cones are an effective way to model partial orders in hyperbolic space, offering physically intuitive and novel insights about the nature of such structures.

This is the compendium of the cluster algebra and quiver package for sage. The purpose of this package is to provide a platform to work with cluster algebras in graduate courses and to further develop the theory by working on examples, by gathering data, and by exhibiting and testing conjectures. In this compendium, we include the relevant theory to introduce the reader to cluster algebras assuming no prior background; this exposition has been written to be accessible to an interested undergraduate. Throughout this compendium, we include examples that the user can run in the sage notebook or command line, and then close with a detailed description of the data structures and methods in this package.

Disentangling the factors of variation in data is a fundamental concept in machine learning and has been studied in various ways by different researchers, leading to a multitude of definitions. Despite the numerous empirical studies, more theoretical research is needed to fully understand the defining properties of disentanglement and how different definitions relate to each other. This paper presents a meta-analysis of existing definitions of disentanglement, using category theory as a unifying and rigorous framework. We propose that the concepts of the cartesian and monoidal products should serve as the core of disentanglement. With these core concepts, we show the similarities and crucial differences in dealing with (i) functions, (ii) equivariant maps, (iii) relations, and (iv) stochastic maps. Overall, our meta-analysis deepens our understanding of disentanglement and its various formulations and can help researchers navigate different definitions and choose the most appropriate one for their specific context.

We prove that the homotopy limits and homotopy colimits of chain complexes can be computed by the cobar and bar constructions. We also show that the totalizations of double complexes compute the homotopy limits and homotopy colimits of simplicial and cosimplicial chain complexes.

This paper proposes a novel framework for understanding large language models (LLMs) by reconceptualizing them as semiotic machines rather than as imitations of human cognition. Drawing from structuralist and post-structuralist theories of language-specifically the works of Ferdinand de Saussure and Jacques Derrida-I argue that LLMs should be understood as models of language itself, aligning with Derrida's concept of 'writing' (l'ecriture). The paper is structured into three parts. First, I lay the theoretical groundwork by explaining how the word2vec embedding algorithm operates within Saussure's framework of language as a relational system of signs. Second, I apply Derrida's critique of Saussure to position 'writing' as the object modeled by LLMs, offering a view of the machine's 'mind' as a statistical approximation of sign behavior. Finally, the third section addresses how modern LLMs reflect post-structuralist notions of unfixed meaning, arguing that the "next token generation" mechanism effectively captures the dynamic nature of meaning. By reconceptualizing LLMs as semiotic machines rather than cognitive models, this framework provides an alternative lens through which to assess the strengths and limitations of LLMs, offering new avenues for future research.

We study how large language models (LLMs) ``think'' through their representation space. We propose a novel geometric framework that models an LLM's reasoning as flows -- embedding trajectories evolving where logic goes. We disentangle logical structure from semantics by employing the same natural deduction propositions with varied semantic carriers, allowing us to test whether LLMs internalize logic beyond surface form. This perspective connects reasoning with geometric quantities such as position, velocity, and curvature, enabling formal analysis in representation and concept spaces. Our theory establishes: (1) LLM reasoning corresponds to smooth flows in representation space, and (2) logical statements act as local controllers of these flows' velocities. Using learned representation proxies, we design controlled experiments to visualize and quantify reasoning flows, providing empirical validation of our theoretical framework. Our work serves as both a conceptual foundation and practical tools for studying reasoning phenomenon, offering a new lens for interpretability and formal analysis of LLMs' behavior.

NVIDIA's CUTLASS library provides a robust and expressive set of methods for describing and manipulating multi-dimensional tensor data on the GPU. These methods are conceptually grounded in the abstract notion of a CuTe layout and a rich algebra of such layouts, including operations such as composition, logical product, and logical division. In this paper, we present a categorical framework for understanding this layout algebra by focusing on a naturally occurring class of tractable layouts. To this end, we define two categories Tuple and Nest whose morphisms give rise to layouts. We define a suite of operations on morphisms in these categories and prove their compatibility with the corresponding layout operations. Moreover, we give a complete characterization of the layouts which arise from our construction. Finally, we provide a Python implementation of our categorical constructions, along with tests that demonstrate alignment with CUTLASS behavior. This implementation can be found at our git repository https://github.com/ColfaxResearch/layout-categories.

The categories of open learners (due to Fong, Spivak and Tuy\'eras) and open games (due to the present author, Ghani, Winschel and Zahn) bear a very striking and unexpected similarity. The purpose of this short note is to prove that there is a faithful symmetric monoidal functor from the former to the latter, which means that any supervised neural network (without feedback or other complicating features) can be seen as an open game in a canonical way. Roughly, each parameter is controlled by a different player, and the game's best response relation encodes the dynamics of gradient descent. We suggest paths for further work exploiting the link.

This is an essay in what might be called ``mathematical metaphysics.'' There is a fundamental duality that run through mathematics and the natural sciences. The duality starts as the logical level; it is represented by the Boolean logic of subsets and the logic of partitions since subsets and partitions are category-theoretic dual concepts. In more basic terms, it starts with the duality between the elements (Its) of subsets and the distinctions (Dits, i.e., ordered pairs of elements in different blocks) of a partition. Mathematically, the Its & Dits duality is fully developed in category theory as the reverse-the-arrows duality. The quantitative versions of subsets and partitions are developed as probability theory and information theory (based on logical entropy). Classical physics was based on a view of reality as definite all the way down. In contrast, quantum physics embodies (objective) indefiniteness. And finally, there are the two fundamental dual mechanisms at work in biology, the selectionist mechanism and the generative mechanism, two mechanisms that embody the fundamental duality.

A compelling portrayal of characters is essential to the success of narrative writing. For readers, appreciating a character's traits requires the ability to infer their evolving beliefs, desires, and intentions over the course of a complex storyline, a cognitive skill known as Theory-of-Mind (ToM). Performing ToM reasoning in prolonged narratives requires readers to integrate historical context with current narrative information, a task at which humans excel but Large Language Models (LLMs) often struggle. To systematically evaluate LLMs' ToM reasoning capability in long narratives, we construct LitCharToM, a benchmark of character-centric questions across four ToM dimensions from classic literature. Further, we introduce EvolvTrip, a perspective-aware temporal knowledge graph that tracks psychological development throughout narratives. Our experiments demonstrate that EvolvTrip consistently enhances performance of LLMs across varying scales, even in challenging extended-context scenarios. EvolvTrip proves to be particularly valuable for smaller models, partially bridging the performance gap with larger LLMs and showing great compatibility with lengthy narratives. Our findings highlight the importance of explicit representation of temporal character mental states in narrative comprehension and offer a foundation for more sophisticated character understanding. Our data and code are publicly available at https://github.com/Bernard-Yang/EvolvTrip.

The Shadowing Principle of Manin has proved a valuable tool for addressing questions of quantitative topology raised by Gromov in the late 1900s. The principle informally provides a way for bounded algebraic maps between differential graded algebras to be translated into nearby genuine maps between their geometric realizations. We extend this principle to finite towers of principal K(G,n) fibrations, and in particular apply this construction to nilpotent spaces. As a specific application of the extended principle, we provide upper bounds on the asymptotic behavior of volumes of nullhomotopies of Lipschitz maps into nilpotent spaces. We further refine these bounds in the case when c = 1 to nearly meet those of the simply connected setting. We similarly refine these bounds in the event the target space is coformal, and demonstrate that the bounds in this setting are nearly sharp.

Large language models (LLMs) are becoming deeply embedded in human communication and decision-making, yet they inherit the ambiguity, bias, and lack of direct access to truth inherent in language itself. While their outputs are fluent, emotionally resonant, and coherent, they are generated through statistical prediction rather than grounded reasoning. This creates the risk of hallucination, responses that sound convincing but lack factual validity. Building on Geoffrey Hinton's observation that AI mirrors human intuition rather than reasoning, this paper argues that LLMs operationalize System 1 cognition at scale: fast, associative, and persuasive, but without reflection or falsification. To address this, we introduce the Rose-Frame, a three-dimensional framework for diagnosing cognitive and epistemic drift in human-AI interaction. The three axes are: (i) Map vs. Territory, which distinguishes representations of reality (epistemology) from reality itself (ontology); (ii) Intuition vs. Reason, drawing on dual-process theory to separate fast, emotional judgments from slow, reflective thinking; and (iii) Conflict vs. Confirmation, which examines whether ideas are critically tested through disagreement or simply reinforced through mutual validation. Each dimension captures a distinct failure mode, and their combination amplifies misalignment. Rose-Frame does not attempt to fix LLMs with more data or rules. Instead, it offers a reflective tool that makes both the model's limitations and the user's assumptions visible, enabling more transparent and critically aware AI deployment. It reframes alignment as cognitive governance: intuition, whether human or artificial, must remain governed by human reason. Only by embedding reflective, falsifiable oversight can we align machine fluency with human understanding.

We present a novel framework for learning on CW-complex structured data points. Recent advances have discussed CW-complexes as ideal learning representations for problems in cheminformatics. However, there is a lack of available machine learning methods suitable for learning on CW-complexes. In this paper we develop notions of convolution and attention that are well defined for CW-complexes. These notions enable us to create the first Hodge informed neural network that can receive a CW-complex as input. We illustrate and interpret this framework in the context of supervised prediction.

In the previous article, we presented a quantum-inspired framework for modeling semantic representation and processing in Large Language Models (LLMs), drawing upon mathematical tools and conceptual analogies from quantum mechanics to offer a new perspective on these complex systems. In this paper, we clarify the core assumptions of this model, providing a detailed exposition of six key principles that govern semantic representation, interaction, and dynamics within LLMs. The goal is to justify that a quantum-inspired framework is a valid approach to studying semantic spaces. This framework offers valuable insights into their information processing and response generation, and we further discuss the potential of leveraging quantum computing to develop significantly more powerful and efficient LLMs based on these principles.

In the rapidly evolving field of Large Language Models (LLMs), there is a critical need to thoroughly analyze their capabilities and risks. Central to our investigation are two novel elements. Firstly, it is the innovative parallels between the statistical patterns of word relationships within LLMs and Martin Heidegger's concepts of "ready-to-hand" and "present-at-hand," which encapsulate the utilitarian and scientific altitudes humans employ in interacting with the world. This comparison lays the groundwork for positioning LLMs as the digital counterpart to the Faculty of Verbal Knowledge, shedding light on their capacity to emulate certain facets of human reasoning. Secondly, a structural analysis of human reasoning, viewed through Heidegger's notion of truth as "unconcealment" is conducted This foundational principle enables us to map out the inputs and outputs of the reasoning system and divide reasoning into four distinct categories. Respective cognitive faculties are delineated, allowing us to place LLMs within the broader schema of human reasoning, thus clarifying their strengths and inherent limitations. Our findings reveal that while LLMs possess the capability for Direct Explicative Reasoning and Pseudo Rational Reasoning, they fall short in authentic rational reasoning and have no creative reasoning capabilities, due to the current lack of many analogous AI models such as the Faculty of Judgement. The potential and risks of LLMs when they are augmented with other AI technologies are also evaluated. The results indicate that although LLMs have achieved proficiency in some reasoning abilities, the aspiration to match or exceed human intellectual capabilities is yet unattained. This research not only enriches our comprehension of LLMs but also propels forward the discourse on AI's potential and its bounds, paving the way for future explorations into AI's evolving landscape.

We present an algorithm to compute the torsion component Pic^τX of the Picard scheme of a smooth projective variety X over a field k. Specifically, we describe Pic^τX as a closed subscheme of a projective space defined by explicit homogeneous polynomials. Furthermore, we compute the group scheme structure on Pic^τX. As applications, we provide algorithms to compute various homological invariants. Among these, we compute the abelianization of the geometric étale fundamental group π^{{et}}_1(X_{k}, x)^{ab}. Moreover, we determine the Galois module structure of the first étale cohomology groups H^1_{{et}}(X_{k}, Z/nZ) without requiring n to be prime to the characteristic of k.

Human perception of the world is shaped by a multitude of viewpoints and modalities. While many existing datasets focus on scene understanding from a certain perspective (e.g. egocentric or third-person views), our dataset offers a panoptic perspective (i.e. multiple viewpoints with multiple data modalities). Specifically, we encapsulate third-person panoramic and front views, as well as egocentric monocular/binocular views with rich modalities including video, multi-channel audio, directional binaural delay, location data and textual scene descriptions within each scene captured, presenting comprehensive observation of the world. Figure 1 offers a glimpse of all 28 scene categories of our 360+x dataset. To the best of our knowledge, this is the first database that covers multiple viewpoints with multiple data modalities to mimic how daily information is accessed in the real world. Through our benchmark analysis, we presented 5 different scene understanding tasks on the proposed 360+x dataset to evaluate the impact and benefit of each data modality and perspective in panoptic scene understanding. We hope this unique dataset could broaden the scope of comprehensive scene understanding and encourage the community to approach these problems from more diverse perspectives.

Narrative comprehension on long stories and novels has been a challenging domain attributed to their intricate plotlines and entangled, often evolving relations among characters and entities. Given the LLM's diminished reasoning over extended context and high computational cost, retrieval-based approaches remain a pivotal role in practice. However, traditional RAG methods can fall short due to their stateless, single-step retrieval process, which often overlooks the dynamic nature of capturing interconnected relations within long-range context. In this work, we propose ComoRAG, holding the principle that narrative reasoning is not a one-shot process, but a dynamic, evolving interplay between new evidence acquisition and past knowledge consolidation, analogous to human cognition when reasoning with memory-related signals in the brain. Specifically, when encountering a reasoning impasse, ComoRAG undergoes iterative reasoning cycles while interacting with a dynamic memory workspace. In each cycle, it generates probing queries to devise new exploratory paths, then integrates the retrieved evidence of new aspects into a global memory pool, thereby supporting the emergence of a coherent context for the query resolution. Across four challenging long-context narrative benchmarks (200K+ tokens), ComoRAG outperforms strong RAG baselines with consistent relative gains up to 11% compared to the strongest baseline. Further analysis reveals that ComoRAG is particularly advantageous for complex queries requiring global comprehension, offering a principled, cognitively motivated paradigm for retrieval-based long context comprehension towards stateful reasoning. Our code is publicly released at https://github.com/EternityJune25/ComoRAG

A key goal of current mechanistic interpretability research in NLP is to find linear features (also called "feature vectors") for transformers: directions in activation space corresponding to concepts that are used by a given model in its computation. Present state-of-the-art methods for finding linear features require large amounts of labelled data -- both laborious to acquire and computationally expensive to utilize. In this work, we introduce a novel method, called "observable propagation" (in short: ObsProp), for finding linear features used by transformer language models in computing a given task -- using almost no data. Our paradigm centers on the concept of observables, linear functionals corresponding to given tasks. We then introduce a mathematical theory for the analysis of feature vectors: we provide theoretical motivation for why LayerNorm nonlinearities do not affect the direction of feature vectors; we also introduce a similarity metric between feature vectors called the coupling coefficient which estimates the degree to which one feature's output correlates with another's. We use ObsProp to perform extensive qualitative investigations into several tasks, including gendered occupational bias, political party prediction, and programming language detection. Our results suggest that ObsProp surpasses traditional approaches for finding feature vectors in the low-data regime, and that ObsProp can be used to better understand the mechanisms responsible for bias in large language models. Code for experiments can be found at github.com/jacobdunefsky/ObservablePropagation.

Mechanistic interpretability aims to understand model behaviors in terms of specific, interpretable features, often hypothesized to manifest as low-dimensional subspaces of activations. Specifically, recent studies have explored subspace interventions (such as activation patching) as a way to simultaneously manipulate model behavior and attribute the features behind it to given subspaces. In this work, we demonstrate that these two aims diverge, potentially leading to an illusory sense of interpretability. Counterintuitively, even if a subspace intervention makes the model's output behave as if the value of a feature was changed, this effect may be achieved by activating a dormant parallel pathway leveraging another subspace that is causally disconnected from model outputs. We demonstrate this phenomenon in a distilled mathematical example, in two real-world domains (the indirect object identification task and factual recall), and present evidence for its prevalence in practice. In the context of factual recall, we further show a link to rank-1 fact editing, providing a mechanistic explanation for previous work observing an inconsistency between fact editing performance and fact localization. However, this does not imply that activation patching of subspaces is intrinsically unfit for interpretability. To contextualize our findings, we also show what a success case looks like in a task (indirect object identification) where prior manual circuit analysis informs an understanding of the location of a feature. We explore the additional evidence needed to argue that a patched subspace is faithful.

Multimodal Large Language Models (MLLMs) show strong visual perception, yet remain limited in reasoning about space under changing viewpoints. We study this challenge as Perspective-Conditioned Spatial Reasoning (PCSR) in 360-degree omnidirectional images, where broad scene coverage reduces ambiguity from partial observations without eliminating the need for viewpoint-dependent inference. To assess this capability, we introduce PCSR-Bench, a diagnostic benchmark of 84,373 question-answer pairs from 2,600 omnidirectional images across 26 indoor environments. PCSR-Bench contains eight tasks spanning foundational perception (e.g., object counting, relative distance, and relative direction) and advanced PCSR, including compositional chains, egocentric rotation, perspective re-anchoring, ego-distortion, and limited-FOV visibility. We evaluate 14 representative MLLMs and observe a substantial perception-reasoning gap: accuracy reaches 57.59% on foundational relative direction, but drops to 13.49% on egocentric rotation, 7.13% on egocentric distortion, and 0.64% on open-ended compositional reasoning. To probe the plasticity of this gap, we conduct an RL-based diagnostic study on a 7B-scale model. Reward shaping improves a matched 7B baseline from 31.10% to 60.06% under a controlled setting, suggesting that PCSR is partial plasticity rather than being fully immutable. Still, the gains are task-selective, sensitive to reward design including both weight allocation and reward formulation, and partially dependent on the evaluation protocol. These results position PCSR as a key bottleneck in current MLLMs and highlight limited but meaningful room for recovery under targeted optimization.

Recent advancements in general-purpose AI have highlighted the importance of guiding AI systems towards the intended goals, ethical principles, and values of individuals and groups, a concept broadly recognized as alignment. However, the lack of clarified definitions and scopes of human-AI alignment poses a significant obstacle, hampering collaborative efforts across research domains to achieve this alignment. In particular, ML- and philosophy-oriented alignment research often views AI alignment as a static, unidirectional process (i.e., aiming to ensure that AI systems' objectives match humans) rather than an ongoing, mutual alignment problem [429]. This perspective largely neglects the long-term interaction and dynamic changes of alignment. To understand these gaps, we introduce a systematic review of over 400 papers published between 2019 and January 2024, spanning multiple domains such as Human-Computer Interaction (HCI), Natural Language Processing (NLP), Machine Learning (ML), and others. We characterize, define and scope human-AI alignment. From this, we present a conceptual framework of "Bidirectional Human-AI Alignment" to organize the literature from a human-centered perspective. This framework encompasses both 1) conventional studies of aligning AI to humans that ensures AI produces the intended outcomes determined by humans, and 2) a proposed concept of aligning humans to AI, which aims to help individuals and society adjust to AI advancements both cognitively and behaviorally. Additionally, we articulate the key findings derived from literature analysis, including discussions about human values, interaction techniques, and evaluations. To pave the way for future studies, we envision three key challenges for future directions and propose examples of potential future solutions.

In this paper, we give a description of the self-injective dimension of string algebras and obtain a necessary and sufficient condition for a string algebra to be Gorenstein.

We define and develop a homotopy invariant notion for the topological complexity of a map f:X to Y, denoted TC(f), that interacts with TC(X) and TC(Y) in the same way cat(f) interacts with cat(X) and cat(Y). Furthermore, TC(f) and cat(f) satisfy the same inequalities as TC(X) and cat(X). We compare it to other invariants defined in the papers [15,16,17,18,20]. We apply TC(f) to studying group homomorphisms f:Hto G.

Given an arbitrary, finitely presented, residually finite group Γ, one can construct a finitely generated, residually finite, free-by-free group M_Γ= F_inftyrtimes F_4 and an embedding M_Γhookrightarrow (F_4ast Γ)times F_4 that induces an isomorphism of profinite completions. In particular, there is a free-by-free group whose profinite completion contains widehatΓ as a retract.

Semantic word embeddings represent the meaning of a word via a vector, and are created by diverse methods. Many use nonlinear operations on co-occurrence statistics, and have hand-tuned hyperparameters and reweighting methods. This paper proposes a new generative model, a dynamic version of the log-linear topic model of~mnih2007three. The methodological novelty is to use the prior to compute closed form expressions for word statistics. This provides a theoretical justification for nonlinear models like PMI, word2vec, and GloVe, as well as some hyperparameter choices. It also helps explain why low-dimensional semantic embeddings contain linear algebraic structure that allows solution of word analogies, as shown by~mikolov2013efficient and many subsequent papers. Experimental support is provided for the generative model assumptions, the most important of which is that latent word vectors are fairly uniformly dispersed in space.

We investigate a construction providing pairs of Calabi-Yau varieties described as zero loci of pushforwards of a hyperplane section on a roof as described by Kanemitsu. We discuss the implications of such construction at the level of Hodge equivalence, derived equivalence and mathbb L-equivalence. For the case of K3 surfaces, we provide alternative interpretations for the Fourier-Mukai duality in the family of K3 surfaces of degree 12 of Mukai. In all these constructions the derived equivalence lifts to an equivalence of matrix factorizations categories.

We propose LENS, a modular approach for tackling computer vision problems by leveraging the power of large language models (LLMs). Our system uses a language model to reason over outputs from a set of independent and highly descriptive vision modules that provide exhaustive information about an image. We evaluate the approach on pure computer vision settings such as zero- and few-shot object recognition, as well as on vision and language problems. LENS can be applied to any off-the-shelf LLM and we find that the LLMs with LENS perform highly competitively with much bigger and much more sophisticated systems, without any multimodal training whatsoever. We open-source our code at https://github.com/ContextualAI/lens and provide an interactive demo.

Capable language models hold what a character believes apart from what is true: told "Anna believes the cup is blue; in reality it is red," they answer blue about Anna and red about the world. Where in the computation does that separation live? We show it rests on two separable mechanisms at two positions. A generic value slot binds the attributed value. A router at the query position selects which frame, the character's belief or reality, a query reads out. Two routes fill the slot: an asserted belief, whose value the text supplies, binds in directly; a derived belief, whose value must be inferred from what the character could see, arrives by a visibility-gated lookback. A subspace trained on either route steers the other, and only the derived route depends on described visibility. The slot itself carries no belief-reality tag: intervening on it moves a reality readout as strongly as a belief one. The separation lives instead in a dissociated pair of routing subspaces, which flip a query between frames without injecting the donor's value. These results hold across three architectures, on stimuli de-confounded against theory-of-mind-benchmark shortcuts; the behavior itself emerges between 3B and 7B across five model families. This paper develops the single belief-reality axis in depth; a companion paper shows the same slot-and-router format is shared across the other non-actual contexts a sentence can open (counterfactual, fictional, temporal).

Linguistic commentary on LLMs, heavily influenced by the theoretical frameworks of de Saussure and Chomsky, is often speculative and unproductive. Critics challenge whether LLMs can legitimately model language, citing the need for "deep structure" or "grounding" to achieve an idealized linguistic "competence." We argue for a radical shift in perspective towards the empiricist principles of Witold Ma\'nczak, a prominent general and historical linguist. He defines language not as a "system of signs" or a "computational system of the brain" but as the totality of all that is said and written. Above all, he identifies frequency of use of particular language elements as language's primary governing principle. Using his framework, we challenge prior critiques of LLMs and provide a constructive guide for designing, evaluating, and interpreting language models.

We extend Textual Inversion to learn pseudo-words that represent a concept at different resolutions. This allows us to generate images that use the concept with different levels of detail and also to manipulate different resolutions using language. Once learned, the user can generate images at different levels of agreement to the original concept; "A photo of S^*(0)" produces the exact object while the prompt "A photo of S^*(0.8)" only matches the rough outlines and colors. Our framework allows us to generate images that use different resolutions of an image (e.g. details, textures, styles) as separate pseudo-words that can be composed in various ways. We open-soure our code in the following URL: https://github.com/giannisdaras/multires_textual_inversion

One of the main methods for computational interpretation of a text is mapping it into a vector in some embedding space. Such vectors can then be used for a variety of textual processing tasks. Recently, most embedding spaces are a product of training large language models (LLMs). One major drawback of this type of representation is their incomprehensibility to humans. Understanding the embedding space is crucial for several important needs, including the need to debug the embedding method and compare it to alternatives, and the need to detect biases hidden in the model. In this paper, we present a novel method of understanding embeddings by transforming a latent embedding space into a comprehensible conceptual space. We present an algorithm for deriving a conceptual space with dynamic on-demand granularity. We devise a new evaluation method, using either human rater or LLM-based raters, to show that the conceptualized vectors indeed represent the semantics of the original latent ones. We show the use of our method for various tasks, including comparing the semantics of alternative models and tracing the layers of the LLM. The code is available online https://github.com/adiSimhi/Interpreting-Embedding-Spaces-by-Conceptualization.

Sparse autoencoders have recently produced dictionaries of high-dimensional vectors corresponding to the universe of concepts represented by large language models. We find that this concept universe has interesting structure at three levels: 1) The "atomic" small-scale structure contains "crystals" whose faces are parallelograms or trapezoids, generalizing well-known examples such as (man-woman-king-queen). We find that the quality of such parallelograms and associated function vectors improves greatly when projecting out global distractor directions such as word length, which is efficiently done with linear discriminant analysis. 2) The "brain" intermediate-scale structure has significant spatial modularity; for example, math and code features form a "lobe" akin to functional lobes seen in neural fMRI images. We quantify the spatial locality of these lobes with multiple metrics and find that clusters of co-occurring features, at coarse enough scale, also cluster together spatially far more than one would expect if feature geometry were random. 3) The "galaxy" scale large-scale structure of the feature point cloud is not isotropic, but instead has a power law of eigenvalues with steepest slope in middle layers. We also quantify how the clustering entropy depends on the layer.

We introduce the specialization map in Scholzes theory of diamonds. We consider v-sheaves that behave like formal schemes and call them kimberlites. We attach to them: a reduced special fiber, an analytic locus, a specialization map, a Zariski site, and an etale site. When the kimberlite comes from a formal scheme, our sites recover the classical ones. We prove that unramified p-adic Beilinson--Drinfeld Grassmannians are kimberlites with finiteness and normality properties.

The learnable, linear neural network layers between tensor power spaces of R^{n} that are equivariant to the orthogonal group, O(n), the special orthogonal group, SO(n), and the symplectic group, Sp(n), were characterised in arXiv:2212.08630. We present an algorithm for multiplying a vector by any weight matrix for each of these groups, using category theoretic constructions to implement the procedure. We achieve a significant reduction in computational cost compared with a naive implementation by making use of Kronecker product matrices to perform the multiplication. We show that our approach extends to the symmetric group, S_n, recovering the algorithm of arXiv:2303.06208 in the process.

Language model representations often contain linear directions that correspond to high-level concepts. Here, we study the dynamics of these representations: how representations evolve along these dimensions within the context of (simulated) conversations. We find that linear representations can change dramatically over a conversation; for example, information that is represented as factual at the beginning of a conversation can be represented as non-factual at the end and vice versa. These changes are content-dependent; while representations of conversation-relevant information may change, generic information is generally preserved. These changes are robust even for dimensions that disentangle factuality from more superficial response patterns, and occur across different model families and layers of the model. These representation changes do not require on-policy conversations; even replaying a conversation script written by an entirely different model can produce similar changes. However, adaptation is much weaker from simply having a sci-fi story in context that is framed more explicitly as such. We also show that steering along a representational direction can have dramatically different effects at different points in a conversation. These results are consistent with the idea that representations may evolve in response to the model playing a particular role that is cued by a conversation. Our findings may pose challenges for interpretability and steering -- in particular, they imply that it may be misleading to use static interpretations of features or directions, or probes that assume a particular range of features consistently corresponds to a particular ground-truth value. However, these types of representational dynamics also point to exciting new research directions for understanding how models adapt to context.

We present SpinBench, a cognitively grounded diagnostic benchmark for evaluating spatial reasoning in vision language models (VLMs). SpinBench is designed around the core challenge of spatial reasoning: perspective taking, the ability to reason about how scenes and object relations change under viewpoint transformation. Since perspective taking requires multiple cognitive capabilities, such as recognizing objects across views, relative positions grounding, and mentally simulating transformations, SpinBench introduces a set of fine-grained diagnostic categories. Our categories target translation, rotation, object relative pose, and viewpoint change, and are progressively structured so that single-object simpler tasks scaffold toward the most demanding multi-object perspective-taking setting. We evaluate 37 state-of-the-art VLMs, both proprietary and open source. Results reveal systematic weaknesses: strong egocentric bias, poor rotational understanding, and inconsistencies under symmetrical and syntactic reformulations. Scaling analysis shows both smooth improvements and emergent capabilities. While human subjects achieve high accuracy (91.2\%), task difficulty as measured by human response time shows strong correlation with VLM accuracy, indicating that SpinBench captures spatial reasoning challenges shared across humans and VLMs. We believe SpinBench provides critical insights into spatial reasoning in VLMs and highlights key gaps in their ability to reason about physical space. Our website can be found at https://spinbench25.github.io/.

We propose a formal reconstruction of quantum mechanics grounded not in external mathematical abstractions, but in the structured dynamics of subjective experience. The Qualia Abstraction Language (QAL) models physical systems as evolving streams of introspective units, structured sequences of modality, shape, and functional effect, rather than as state vectors in Hilbert space. This approach reimagines core quantum concepts: superposition becomes a form of structured ambiguity; collapse is reframed as an introspective contraction; and entanglement is modeled as semantic resonance across streams of qualia. Drawing on insights from nominalist philosophy and oversight theoretic limits in AI, we argue that the observer paradox in quantum mechanics reflects not an ontological lacuna, but a linguistic one: the absence of a formal vocabulary for modeling first person structure. QAL introduces such a vocabulary, providing a morphodynamic framework that embeds the observer within the system and replaces abstract projection with endogenous transformation. We analyze the alignment of QAL with endophysical approaches, contrast it with standard interpretations of quantum theory, and explore its implications for a post Platonist, introspectively grounded physics.

Does structure in representations imply structure in computation? We study how Llama-3.1-8B reasons over cyclic concepts (e.g., "what month is six months after August?"). Even though Llama-3.1-8B's representations for these concepts are circularly structured, we find that instead of directly computing modular addition in the period of the cyclic concept (e.g., 12 for months), the model re-uses a generic addition mechanism across tasks that operates independently of concept-specific geometry. First, it computes the sum of its two inputs using base-10 addition (six + August=14). Then, it maps this sum back to cyclic concept space (14->February). We show that Llama-3.1-8B uses task-agnostic Fourier features to compute these sums--in fact, these features have periods that respect standard base-10 addition, e.g., 2, 5, and 10, rather than the cyclic concept period (e.g., 12 for months). Furthermore, we identify a sparse set of 28 MLP neurons re-used across all tasks (approximately 0.2% of the MLP at layer 18) that can be partitioned into disjoint clusters, each computing the sum for a Fourier feature with a different period. Our work highlights how an interplay between causal abstraction and feature geometry can deepen our mechanistic understanding of LMs.

Assessing factuality of text generated by large language models (LLMs) is an emerging yet crucial research area, aimed at alerting users to potential errors and guiding the development of more reliable LLMs. Nonetheless, the evaluators assessing factuality necessitate suitable evaluation themselves to gauge progress and foster advancements. This direction remains under-explored, resulting in substantial impediments to the progress of factuality evaluators. To mitigate this issue, we introduce a benchmark for Factuality Evaluation of large Language Models, referred to as felm. In this benchmark, we collect responses generated from LLMs and annotate factuality labels in a fine-grained manner. Contrary to previous studies that primarily concentrate on the factuality of world knowledge (e.g.~information from Wikipedia), felm focuses on factuality across diverse domains, spanning from world knowledge to math and reasoning. Our annotation is based on text segments, which can help pinpoint specific factual errors. The factuality annotations are further supplemented by predefined error types and reference links that either support or contradict the statement. In our experiments, we investigate the performance of several LLM-based factuality evaluators on felm, including both vanilla LLMs and those augmented with retrieval mechanisms and chain-of-thought processes. Our findings reveal that while retrieval aids factuality evaluation, current LLMs are far from satisfactory to faithfully detect factual errors.