# Categories of Differentiable Polynomial Circuits for Machine Learning

arXiv: https://arxiv.org/abs/2203.06430 · HTML: https://mlsanity.com/paper/2203.06430/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 10 of its 31 references that are in the map.
- First posted on arXiv 12 Mar 2022.

## Builds on (most cited first, 10 of 10)

- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Estimating or Propagating Gradients Through Stochastic Neurons for Conditional Computation](https://mlsanity.com/paper/1308.3432/) (2013)
- [Categorical Foundations of Gradient-Based Learning](https://mlsanity.com/paper/2103.01931/) (2021)
- [Reverse derivative categories](https://mlsanity.com/paper/1910.07065/) (2019)
- [Reverse Derivative Ascent: A Categorical Approach to Learning Boolean Circuits](https://mlsanity.com/paper/2101.10488/) (2021)
- [Interacting Hopf Algebras: the theory of linear systems](https://mlsanity.com/paper/1805.03032/) (2018)
- [String Diagram Rewrite Theory I: Rewriting with Frobenius Structure](https://mlsanity.com/paper/2012.01847/) (2020)
- [String Diagram Rewrite Theory II: Rewriting with Symmetric Monoidal Structure](https://mlsanity.com/paper/2104.14686/) (2021)
- [The Cost of Compositionality: A High-Performance Implementation of String Diagram Composition](https://mlsanity.com/paper/2105.09257/) (2021)
- [String Diagram Rewrite Theory III: Confluence with and without Frobenius](https://mlsanity.com/paper/2109.06049/) (2021)

## Built on by (most cited first, 4 of 4)

- [Neural Circuit Diagrams: Robust Diagrams for the Communication, Implementation, and Analysis of Deep Learning Architectures](https://mlsanity.com/paper/2402.05424/) (2024)
- [Deep Learning with Parametric Lenses](https://mlsanity.com/paper/2404.00408/) (2024)
- [Accelerating Machine Learning Systems via Category Theory: Applications to Spherical Attention for Gene Regulatory Networks](https://mlsanity.com/paper/2505.09326/) (2025)
- [Reverse Fa`a di Bruno's Formula for Cartesian Reverse Differential Categories](https://mlsanity.com/paper/2509.20931/) (2025)

## Abstract (arXiv)

Reverse derivative categories (RDCs) have recently been shown to be a suitable semantic framework for studying machine learning algorithms. Whereas emphasis has been put on training methodologies, less attention has been devoted to particular model classes: the concrete categories whose morphisms represent machine learning models. In this paper we study presentations by generators and equations of classes of RDCs. In particular, we propose polynomial circuits as a suitable machine learning model. We give an axiomatisation for these circuits and prove a functional completeness result. Finally, we discuss the use of polynomial circuits over specific semirings to perform machine learning with discrete values.