# Category Theory in Machine Learning

arXiv: https://arxiv.org/abs/2106.07032 · HTML: https://mlsanity.com/paper/2106.07032/

## Key facts

- Cited by 36 papers in the map (data through 14 Oct 2026).
- Builds on 38 of its 108 references that are in the map.
- First posted on arXiv 13 Jun 2021.

## Builds on (most cited first, 30 of 38)

- [Language Models are Few-Shot Learners](https://mlsanity.com/paper/2005.14165/) (2020)
- [UMAP: Uniform Manifold Approximation and Projection for Dimension Reduction](https://mlsanity.com/paper/1802.03426/) (2018)
- [Group Equivariant Convolutional Networks](https://mlsanity.com/paper/1602.07576/) (2016)
- [Geometric deep learning: going beyond Euclidean data](https://mlsanity.com/paper/1611.08097/) (2016)
- [An overview of gradient descent optimization algorithms](https://mlsanity.com/paper/1609.04747/) (2016)
- [Spherical CNNs](https://mlsanity.com/paper/1801.10130/) (2018)
- [Invariant and Equivariant Graph Networks](https://mlsanity.com/paper/1812.09902/) (2018)
- [On the Generalization of Equivariance and Convolution in Neural Networks to the Action of Compact Groups](https://mlsanity.com/paper/1802.03690/) (2018)
- [Gauge Equivariant Convolutional Networks and the Icosahedral CNN](https://mlsanity.com/paper/1902.04615/) (2019)
- [Graph Neural Network for Traffic Forecasting: A Survey](https://mlsanity.com/paper/2101.11174/) (2021)
- [A General Theory of Equivariant CNNs on Homogeneous Spaces](https://mlsanity.com/paper/1811.02017/) (2018)
- [An introduction to Topological Data Analysis: fundamental and practical aspects for data scientists](https://mlsanity.com/paper/1710.04019/) (2017)
- [A synthetic approach to Markov kernels, conditional independence and theorems on sufficient statistics](https://mlsanity.com/paper/1908.07021/) (2019)
- [Backprop as Functor: A compositional perspective on supervised learning](https://mlsanity.com/paper/1711.10455/) (2017)
- [Categorical Foundations of Gradient-Based Learning](https://mlsanity.com/paper/2103.01931/) (2021)
- [Disintegration and Bayesian Inversion via String Diagrams](https://mlsanity.com/paper/1709.00322/) (2017)
- [The simple essence of automatic differentiation](https://mlsanity.com/paper/1804.00746/) (2018)
- [Interactive Visualization of 2-D Persistence Modules](https://mlsanity.com/paper/1512.00180/) (2015)
- [A Convenient Category for Higher-Order Probability Theory](https://mlsanity.com/paper/1701.02547/) (2017)
- [Reverse derivative categories](https://mlsanity.com/paper/1910.07065/) (2019)
- [Lenses and Learners](https://mlsanity.com/paper/1903.03671/) (2019)
- [Representable Markov Categories and Comparison of Statistical Experiments in Categorical Probability](https://mlsanity.com/paper/2010.07416/) (2020)
- [Reverse Derivative Ascent: A Categorical Approach to Learning Boolean Circuits](https://mlsanity.com/paper/2101.10488/) (2021)
- [On Universal Approximation by Neural Networks with Uniform Guarantees on Approximation of Infinite Dimensional Maps](https://mlsanity.com/paper/1910.01545/) (2019)
- [Compositional Deep Learning](https://mlsanity.com/paper/1907.08292/) (2019)
- [A Channel-Based Perspective on Conjugate Priors](https://mlsanity.com/paper/1707.00269/) (2017)
- [Categorical Stochastic Processes and Likelihood](https://mlsanity.com/paper/2005.04735/) (2020)
- [Learning Functors using Gradient Descent](https://mlsanity.com/paper/2009.06837/) (2020)
- [Differentiable Causal Computations via Delayed Trace](https://mlsanity.com/paper/1903.01093/) (2019)
- [Limits of bimorphic lenses](https://mlsanity.com/paper/1808.05545/) (2018)

## Built on by (most cited first, 30 of 36)

- [On the Power of Foundation Models](https://mlsanity.com/paper/2211.16327/) (2022)
- [Causal models in string diagrams](https://mlsanity.com/paper/2304.07638/) (2023)
- [Going Beyond Neural Network Feature Similarity: The Network Feature Complexity and Its Interpretation Using Category Theory](https://mlsanity.com/paper/2310.06756/) (2023)
- [Towards Compositional Interpretability for XAI](https://mlsanity.com/paper/2406.17583/) (2024)
- [Compositionality as we see it, everywhere around us](https://mlsanity.com/paper/2110.05327/) (2021)
- [Position: LLMs Must Use Functor-Based and RAG-Driven Bias Mitigation for Fairness](https://mlsanity.com/paper/2603.07368/) (2026)
- [CatRAG: Functor-Guided Structural Debiasing with Retrieval Augmentation for Fair LLMs](https://mlsanity.com/paper/2603.21524/) (2026)
- [Neural Circuit Diagrams: Robust Diagrams for the Communication, Implementation, and Analysis of Deep Learning Architectures](https://mlsanity.com/paper/2402.05424/) (2024)
- [Categorical Foundations of Explainable AI: A Unifying Theory](https://mlsanity.com/paper/2304.14094/) (2023)
- [On Meta-Prompting](https://mlsanity.com/paper/2312.06562/) (2023)
- [Pooling Image Datasets With Multiple Covariate Shift and Imbalance](https://mlsanity.com/paper/2403.02598/) (2024)
- [Kan Extensions in Data Science and Machine Learning](https://mlsanity.com/paper/2203.09018/) (2022)
- [A Category-theoretical Meta-analysis of Definitions of Disentanglement](https://mlsanity.com/paper/2305.06886/) (2023)
- [Active Inference in String Diagrams: A Categorical Account of Predictive Processing and Free Energy](https://mlsanity.com/paper/2308.00861/) (2023)
- [From Conceptual Spaces to Quantum Concepts: Formalising and Learning Structured Conceptual Models](https://mlsanity.com/paper/2401.08585/) (2023)
- [Computing with Categories in Machine Learning](https://mlsanity.com/paper/2303.04156/) (2023)
- [Categorification of Group Equivariant Neural Networks](https://mlsanity.com/paper/2304.14144/) (2023)
- [Enriching Disentanglement: From Logical Definitions to Quantitative Metrics](https://mlsanity.com/paper/2305.11512/) (2023)
- [Transport of Algebraic Structure to Latent Embeddings](https://mlsanity.com/paper/2405.16763/) (2024)
- [Towards a Categorical Foundation of Deep Learning: A Survey](https://mlsanity.com/paper/2410.05353/) (2024)
- [An Algebraic Approach to Moralisation and Triangulation of Probabilistic Graphical Models](https://mlsanity.com/paper/2503.11820/) (2025)
- [Logic Explanation of AI Classifiers by Categorical Explaining Functors](https://mlsanity.com/paper/2503.16203/) (2025)
- [From Mechanistic to Compositional Interpretability](https://mlsanity.com/paper/2605.08934/) (2026)
- [A Probabilistic Generative Model of Free Categories](https://mlsanity.com/paper/2205.04545/) (2022)
- [Formalising and Learning a Quantum Model of Concepts](https://mlsanity.com/paper/2302.14822/) (2023)
- [Algebraic Dynamical Systems in Machine Learning](https://mlsanity.com/paper/2311.03118/) (2023)
- [Towards an Algebraic Framework For Approximating Functions Using Neural Network Polynomials](https://mlsanity.com/paper/2402.01058/) (2024)
- [Symmetry-Enriched Learning: A Category-Theoretic Framework for Robust Machine Learning Models](https://mlsanity.com/paper/2409.12100/) (2024)
- [Developing a Foundation of Vector Symbolic Architectures Using Category Theory](https://mlsanity.com/paper/2501.05368/) (2025)
- [The Gauss-Markov Adjunction Provides Categorical Semantics of Residuals in Supervised Learning](https://mlsanity.com/paper/2507.02442/) (2025)

## Abstract (arXiv)

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.