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Category_Theory_Machine_Learning

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List of papers studying machine learning through the lens of category theory

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# Category Theory ∩ Machine Learning

Category theory has been finding increasing applications in machine learning. This repository aims to list all of the relevant papers, grouped by fields.

For an introduction to the ideas behind category theory, check out [this link](https://github.com/bgavran/Category_Theory_Resources).

There might be papers missing, and some papers are in multiple fields. Feel free to contribute to this list or suggest changes to it by creating a pull request or opening an issue.

### Theses

* [Fundamental Components of Deep Learning: A category-theoretic approach](https://arxiv.org/abs/2403.13001)
* [Robust Diagrams for Deep Learning Architectures: Applications and Theory](https://www.vtabbott.io/content/files/2023/11/Robust-Diagrams-for-Deep-Learning-Architectures.pdf)
* [Category-Theoretic Datastructures and Algorithms for Learning Polynomial Circuits](https://eprints.soton.ac.uk/483757/1/paul_wilson_thesis_acrobat_fixup.pdf)
* [Category Theory for Quantum Natural Language Processing](https://arxiv.org/abs/2212.06615)
* [Towards a Categorical Foundation of Deep Learning: A Survey](https://arxiv.org/abs/2410.05353)
  
---

### General Deep Learning
* [Categorical Foundations of Gradient-Based Learning](https://arxiv.org/abs/2103.01931)
* [Position: Categorical Deep Learning is an Algebraic Theory of All Architectures](https://arxiv.org/abs/2402.15332)
* [Coalgebras for categorical deep learning: Representability and universal approximation](https://arxiv.org/abs/2603.03227)
* [Position: Topological Deep Learning is the New Frontier for Relational Learning](https://arxiv.org/abs/2402.08871)
* [Backprop as Functor](https://arxiv.org/abs/1711.10455)
* [Lenses and Learners](https://arxiv.org/abs/1903.03671)
* [Reverse Derivative Ascent](https://arxiv.org/abs/2101.10488)
* [Dioptics](http://events.cs.bham.ac.uk/syco/strings3-syco5/papers/dalrymple.pdf)
* [Learning Functors using Gradient Descent](http://www.cs.ox.ac.uk/ACT2019/preproceedings/Bruno%20Gavranovic.pdf) (longer version [here](https://arxiv.org/abs/1907.08292))
* [Compositionality for Recursive Neural Networks](https://arxiv.org/abs/1901.10723)
* [Deep neural networks as nested dynamical systems](https://arxiv.org/abs/2111.01297)
* [Category Theory in Machine Learning](https://arxiv.org/abs/2106.07032)
* [Neural network layers as parametric spans](https://arxiv.org/abs/2208.00809)
* [Categories of Differentiable Polynomial Circuits for Machine Learning](https://arxiv.org/abs/2203.06430)
* [Learners are almost Free Compact Closed](https://oxford24.github.io/assets/act-papers/39_learners_are_almost_free_compa.pdf)
* [Going Beyond Neural Network Feature Similarity: The Network Feature Complexity and Its Interpretation Using Category Theory](https://arxiv.org/abs/2310.06756)
* [Attending to Topological Spaces: The Cellular Transformer](https://arxiv.org/abs/2405.14094)
* [On the Anatomy of Attention](https://arxiv.org/abs/2407.02423)
* [Algebraic Dynamical Systems in Machine Learning](https://arxiv.org/abs/2311.03118)
* [Can neural operators always be continuously discretized?](https://arxiv.org/abs/2412.03393)
* [Order Theory in the Context of Machine Learning: an application](https://arxiv.org/abs/2412.06097)
* [An Invitation to Neuroalgebraic Geometry](https://arxiv.org/abs/2501.18915)
* [Algebraic Positional Encodings](https://arxiv.org/abs/2312.16045)
* [Learning Structure-Aware Representations of Dependent Types](https://arxiv.org/abs/2402.02104)
* [Accelerating Machine Learning Systems via Category Theory: Applications to Spherical Attention for Gene Regulatory Networks](https://arxiv.org/abs/2505.09326)
* [Copresheaf Topological Neural Networks: A Generalized Deep Learning Framework](https://arxiv.org/abs/2505.21251)
* [Categorical Invariants of Learning Dynamics](https://arxiv.org/abs/2510.04376)
* [Product Interaction: An Algebraic Formalism for Deep Learning Architectures](https://arxiv.org/abs/2602.02573)
* [Presenting Neural Networks via Coherent Functors](https://arxiv.org/abs/2604.15100)
* [Operads for compositional reasoning in LLMs](https://arxiv.org/abs/2606.13634)
* [Learning Transfers: Kan Extensions for Neural Invariants](https://arxiv.org/abs/2606.07627)

#### Equivariance
* [Equivariant neural networks and piecewise linear representation theory](https://arxiv.org/abs/2408.00949)
* [Local Permutation Equivariance For Graph Neural Networks](https://arxiv.org/abs/2111.11840)
* [Stochastic Neural Network Symmetrisation in Markov Categories](https://arxiv.org/abs/2406.11814)
* [Metric Learning for Clifford Group Equivariant Neural Networks](https://arxiv.org/abs/2407.09926)
* [Categorification of Group Equivariant Neural Networks](https://arxiv.org/abs/2304.14144)
* [Equivariant Single View Pose Prediction Via Induced and Restricted Representations](https://arxiv.org/abs/2307.03704)
* [Characterizing the invariances of learning algorithms using category theory](https://arxiv.org/abs/1905.02072)
* [Mathematical Foundation of Interpretable Equivariant Surrogate Models](https://arxiv.org/abs/2503.01942)
* [Filter Equivariant Functions: A symmetric account of length-general extrapolation on lists](https://arxiv.org/abs/2507.08796)
* [Relational inductive biases on attention mechanisms](https://arxiv.org/abs/2507.04117)
* [Identifiable Equivariant Networks are Layerwise Equivariant](https://arxiv.org/abs/2601.21645)
* [Gaussian Sheaf Neural Networks](https://arxiv.org/abs/2605.21435)
* [Theoretical Aspects of Lie Groupoid and Lie Algebroid Equivariant Convolutional Neural Networks](https://arxiv.org/abs/2606.02758)

#### Graph Neural Networks
* [Graph Neural Networks are Dynamic Programmers](https://arxiv.org/abs/2203.15544)
* [Natural Graph Networks](https://arxiv.org/abs/2007.08349)
* [Sheaf Representation Learning](https://www.proquest.com/openview/5ab4860c57313d5f1a02fbe90a505a7f/1?pq-origsite=gscholar&cbl=18750&diss=y)
* [Sheaf Neural Networks](https://arxiv.org/abs/2012.06333)
* [Sheaf Neural Networks with Connection Laplacians](https://arxiv.org/abs/2206.08702)
* [Neural Sheaf Diffusion: A Topological Perspective on Heterophily and Oversmoothing in GNNs](https://arxiv.org/abs/2202.04579)
* [Nonlinear Sheaf Diffusion in Graph Neural Networks](https://arxiv.org/abs/2403.00337)
* [Graph Convolutional Neural Networks as Parametric CoKleisli morphisms](https://arxiv.org/abs/2212.00542)
* [Learnable Commutative Monoids for Graph Neural Networks](https://arxiv.org/abs/2212.08541)
* [Sheaf Neural Networks for Graph-based Recommender Systems](https://arxiv.org/abs/2304.09097)
* [Sheaf theory: from deep geometry to deep learning](https://arxiv.org/abs/2502.15476)
* [Asynchronous Algorithmic Alignment with Cocycles](https://arxiv.org/abs/2306.15632)
* [Topologically Attributed Graphs for Shape Discrimination](https://arxiv.org/abs/2306.17805)
* [Grothendieck Graph Neural Networks Framework: An Algebraic Platform for Crafting Topology-Aware GNNs](https://arxiv.org/abs/2412.08835)
* [Don't be Afraid of Cell Complexes! An Introduction from an Applied Perspective](https://arxiv.org/abs/2506.09726)
* [Graph Lineages and Skeletal Graph Products](https://arxiv.org/abs/2508.00197)
* [On the Sheafification of Higher-Order Message Passing](https://arxiv.org/abs/2509.23020)
* [Disentangling Hyperedges through the Lens of Category Theory](https://arxiv.org/abs/2510.16289)
* [Weisfeiler and Lehman Go Categorical](https://arxiv.org/abs/2602.06787)
* [Modeling Topological Impact on Node Attribute Distributions in Attributed Graphs](https://arxiv.org/abs/2602.01454)

---

### Differentiable programming / automatic differentiation
* [CHAD: Combinatory Homomorphic Automatic Differentiation](https://arxiv.org/abs/2103.15776)
* [CHAD for Expressive Total Languages](https://arxiv.org/abs/2110.00446)
* [The Differentiable Curry](https://dimitriv.github.io/papers/hoad-workshop.pdf)
* [Functorial String Diagrams for Reverse-Mode Automatic Differentiation](https://arxiv.org/abs/2107.13433)
* [Differentiable Causal Computations via Delayed Trace](https://arxiv.org/abs/1903.01093)
* [Simple Essence of Automatic Differentiation](https://arxiv.org/abs/1804.00746)
* [Reverse Derivative Categories](https://arxiv.org/abs/1910.07065)
* [Reverse Faà di Bruno's Formula for Cartesian Reverse Differential Categories](https://arxiv.org/abs/2509.20931)
* [Towards formalizing and extending differential programming using tangent categories](http://www.cs.ox.ac.uk/ACT2019/preproceedings/Jonathan%20Gallagher,%20Geoff%20Cruttwell%20and%20Ben%20MacAdam.pdf)
* [Correctness of Automatic Differentiation via Diffeologies and Categorical Gluing](https://arxiv.org/abs/2001.02209)
* [Denotationally Correct, Purely Functional, Efficient Reverse-mode Automatic Differentiation](https://arxiv.org/abs/2212.09801)
* [Higher Order Automatic Differentiation of Higher Order Functions](https://arxiv.org/abs/2101.06757)
* [Space-time tradeoffs of lenses and optics via higher category theory](https://arxiv.org/abs/2209.09351)
* [Using Rewrite Strategies for Efficient Functional Automatic Differentiation](https://arxiv.org/abs/2307.02447)

### CuTe Layouts

* [CuTe Layout Representation and Algebra](https://arxiv.org/abs/2603.02298v1)
* [Categorical Foundations for CuTe Layouts](https://arxiv.org/abs/2601.05972)

---

### Probability theory
* [Markov categories](https://arxiv.org/abs/1908.07021)
* [Markov Categories and Entropy](https://arxiv.org/abs/2212.11719)
* [Infinite products and zero-one laws in categorical probability](https://arxiv.org/abs/1912.02769)
* [A Convenient Category for Higher-Order Probability Theory](https://arxiv.org/abs/1701.02547)
* [Bimonoidal Structure of Probability Monads](https://arxiv.org/abs/1804.03527)
* [Representable Markov Categories and Comparison of Statistical Experiments in Categorical Probability](https://arxiv.org/abs/2010.07416)
* [De Finneti's construction as a categorical limit](https://arxiv.org/abs/2003.01964)
* [A Probability Monad as the Colimit of Spaces of Finite Samples](https://arxiv.org/abs/1712.05363)
* [A Probabilistic Dependent Type System based on Non-Deterministic Beta Reduction](https://arxiv.org/abs/1602.06420)
* [Probability, valuations, hyperspace: Three monads on Top and the support as a morphism](https://arxiv.org/abs/1910.03752)
* [Categorical Probability Theory](https://arxiv.org/abs/1406.6030)
* [Information structures and their cohomology](https://arxiv.org/abs/1709.07807)
* [Computable Stochastic Processes](https://arxiv.org/abs/1409.4667)
* [Compositional Semantics for Probabilistic Programs with Exact Conditioning](https://arxiv.org/abs/2101.11351)
* [String Diagrams with Factorized Densities](https://arxiv.org/abs/2305.02506)
* [Partial Markov Categories](https://arxiv.org/abs/2502.03477)
* [Random Variables, Conditional Independence and Categories of Abstract Sample Spaces](https://arxiv.org/abs/2503.02477)
* [A categorical account of the Metropolis-Hastings algorithm](https://arxiv.org/abs/2601.22911)

---

### Bayesian/Causal inference
* [The Compositional Structure of Bayesian Inference](https://arxiv.org/abs/2305.06112)
* [Compositional Inference for Bayesian Networks and Causality](https://arxiv.org/abs/2512.00209)
* [Dependent Bayesian Lenses: Categories of Bidirectional Markov Kernels with Canonical Bayesian Inversion](https://arxiv.org/abs/2209.14728)
* [A category theory framework for Bayesian Learning](https://arxiv.org/abs/2111.14293)
* [Causal Theories: A Categorical Perspective on Bayesian Networks](https://arxiv.org/abs/1301.6201)
* [Bayesian machine learning via category theory](https://arxiv.org/abs/1312.1445)
* [A Categorical Foundation for Bayesian probability](https://arxiv.org/abs/1205.1488)
* [Bayesian Open Games](https://arxiv.org/abs/1910.03656)
* [Causal Inference by String Diagram Surgery](https://arxiv.org/abs/1811.08338)
* [Disintegration and Bayesian Inversion via String Diagrams](https://arxiv.org/abs/1709.00322)
* [Categorical Stochastic Processes and Likelihood](https://arxiv.org/abs/2005.04735)
* [Bayesian Updates Compose Optically](https://arxiv.org/abs/2006.01631)
* [Automatic Backward Filtering Forward Guiding for Markov processes and graphical models](https://arxiv.org/abs/2010.03509)
* [Compositionality in algorithms for smoothing](https://arxiv.org/abs/2303.13865)
* [A Channel-Based Perspective on Conjugate Priors](https://arxiv.org/abs/1707.00269)
* [A Type Theory for Probabilistic and Bayesian Reasoning](https://arxiv.org/abs/1511.09230)
* [Denotational validation of higher-order Bayesian inference](https://arxiv.org/abs/1711.03219)
* [The Geometry of Bayesian Programming](https://arxiv.org/abs/1904.07425)
* [Relational Reasoning for Markov Chains in a Probabilistic Guarded Lambda Calculus](https://arxiv.org/abs/1802.09787)
* [A Bayesian Interpretation of the Internal Model Principle](https://arxiv.org/abs/2503.00511)
* [On the Functoriality of Belief Propagation Algorithms on Finite Partially Ordered Sets](https://hal.sorbonne-universite.fr/hal-04995548/document)
* [Bayesian Networks, Markov Networks, Moralisation, Triangulation: a Categorical Perspective](https://arxiv.org/abs/2512.09908)

---

### Topological Data Analysis
* [On Characterizing the Capacity of Neural Networks using Algebraic Topology](https://arxiv.org/abs/1802.04443)
* [Persistent-Homology-based Machine Learning and its Applications - A Survey](https://arxiv.org/abs/1811.00252)
* [Topological Expressiveness of Neural Networks](https://run.unl.pt/bitstream/10362/129615/1/TAA0115.pdf)

---

### Reinforcement Learning

* [Value Iteration is Optic Composition](https://arxiv.org/abs/2206.04547)
* [Compositional Behavioral Semantics for State Abstraction in Reinforcement Learning](https://arxiv.org/abs/2606.25357)
* [Categorical semantics of compositional reinforcement learning](https://arxiv.org/abs/2208.13687)
* [Reduce, Reuse, Recycle: Categories for Compositional Reinforcement Learning](https://arxiv.org/abs/2408.13376v3)

### Metric space magnitude
* [Approximating the convex hull via metric space magnitude](https://arxiv.org/abs/1908.02692)
* [Practical applications of metric space magnitude and weighting vectors](https://arxiv.org/abs/2006.14063)
* [Weighting vectors for machine learning: numerical harmonic analysis applied to boundary detection](https://arxiv.org/abs/2106.00827)
* [The magnitude vector of images](https://arxiv.org/abs/2110.15188)
* [Magnitude of arithmetic scalar and matrix categories](https://arxiv.org/abs/2304.08334)
* [Metric Space Magnitude for Evaluating Unsupervised Representation Learning](https://arxiv.org/abs/2311.16054)
* [The Magnitude of Categories of Texts Enriched by Language Models](https://arxiv.org/abs/2501.06662)

---

### Blog posts
* [Neural Networks, Types, and Functional Programming](https://colah.github.io/posts/2015-09-NN-Types-FP/)
* [Generalized Transformers from Applicative Functors](https://glaive-research.org/2025/02/11/Generalized-Transformers-from-Applicative-Functors.html)
* [Towards Categorical Foundations of Learning](https://www.brunogavranovic.com/posts/2021-03-03-Towards-Categorical-Foundations-Of-Neural-Networks.html)
* [Graph Convolutional Neural Networks as Parametric CoKleisli morphisms](https://www.brunogavranovic.com/posts/2022-12-05-graph_neural_networks_as_parametric_cokleisli_morphisms.html)
* [Optics vs Lenses, Operationally](https://www.brunogavranovic.com/posts/2022-02-10-optics-vs-lenses-operationally.html)
* [Meta-learning and Monads](https://www.brunogavranovic.com/posts/2021-10-13-meta-learning-and-monads.html)

---

### Automata Learning
* [Automata Learning: A Categorical Perspective](http://www.calf-project.org/publications/prakash.pdf)
* [A Categorical Framework for Learning Generalised Tree Automata](https://arxiv.org/abs/2001.05786)
* [CALF: Categorical Automata Learning Framework (thesis)](https://discovery.ucl.ac.uk/id/eprint/10110356/1/thesis_final_ucl.pdf)

---

### Misc
* [Generalized Convolution and Efficient Language Recognition](https://arxiv.org/abs/1903.10677)
* [General supervised learning as change propagation with delta lenses](https://arxiv.org/abs/1911.12904)
* [From Open Learners to Open Games](https://arxiv.org/abs/1902.08666)
* [Learners Languages](https://arxiv.org/abs/2103.01189)
* [A Constructive, Type-Theoretic Approach to Regression via Global Optimisation](https://arxiv.org/abs/2006.12868)
* [Functorial Manifold Learning](https://arxiv.org/abs/2011.07435)
* [Diegetic representation of feedback in open games](https://arxiv.org/abs/2206.12338)
* [Assessing the Unitary RNN as an End-to-End Compositional Model of Syntax](https://arxiv.org/abs/2208.05719)
* [Classifying Clustering Schemes](https://arxiv.org/abs/1011.5270)
* [Categorical Hopfield Networks](https://arxiv.org/abs/2201.02756)
* [A Category-theoretical Meta-analysis of Definitions of Disentanglement](https://arxiv.org/abs/2305.06886)
* [Isomorphism, Normalizing Flows, and Density Estimation: Preserving Relationships Between Data](https://www.cs.uoregon.edu/Reports/AREA-202307-Walton.pdf)
* [Transport of Algebraic Structure to Latent Embeddings](https://brendon-anderson.github.io/files/publications/pfrommer2024transport.pdf)
* [Data for Mathematical Copilots: Better Ways of Presenting Proofs for Machine Learning](https://arxiv.org/abs/2412.15184)
* [Categorical Diffusion of Weighted Lattices](https://arxiv.org/abs/2501.03890)
* [Aggregating time-series and image data: functors and double functors](https://arxiv.org/abs/2504.05274)
* [Logic Explanation of AI Classifiers by Categorical Explaining Functors](https://arxiv.org/abs/2503.16203)
* [Typing tensor calculus in 2-categories](https://arxiv.org/abs/1908.01212v4)
* [The Joys of Categorical Conformal Prediction](https://arxiv.org/abs/2507.04441)
* [The Gauss-Markov Adjunction: Categorical Semantics of Residuals in Supervised Learning](https://arxiv.org/abs/2507.02442)
* [Modeling Layout Abstractions Using Integer Set Relations](https://arxiv.org/abs/2511.10374)
* [Adequate Losses via Quantitative Linear Logic](https://arxiv.org/abs/2605.13348)

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