# Characterizing the invariances of learning algorithms using category theory

arXiv: https://arxiv.org/abs/1905.02072 · HTML: https://mlsanity.com/paper/1905.02072/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 6 May 2019.

## Built on by (most cited first, 3 of 3)

- [Category Theory in Machine Learning](https://mlsanity.com/paper/2106.07032/) (2021)
- [Going Beyond Neural Network Feature Similarity: The Network Feature Complexity and Its Interpretation Using Category Theory](https://mlsanity.com/paper/2310.06756/) (2023)
- [From Intent to Evidence: A Categorical Approach for Structural Evaluation of Deep Research Agents](https://mlsanity.com/paper/2603.25342/) (2026)

## Abstract (arXiv)

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.