Categorical Representation Learning: Morphism is All You Need

TL;DR

This paper introduces a category-theoretic framework for representation learning, called 'categorifier', which models objects and morphisms as vectors and matrices, and demonstrates its effectiveness with a text translation example outperforming traditional models.

Contribution

It develops a novel category-theoretic approach to representation learning, extending beyond set-theoretic methods, and provides a proof-of-concept application in text translation.

Findings

Categorical approach outperforms deep learning models by 17 times in text translation

Objects and morphisms are represented as vectors and matrices in the framework

Framework is part of a US patent proposal

Abstract

We provide a construction for categorical representation learning and introduce the foundations of "$\textit{categorifier}$". The central theme in representation learning is the idea of $\textbf{everything to vector}$. Every object in a dataset $\mathcal{S}$ can be represented as a vector in $\mathbb{R}^n$ by an $\textit{encoding map}$ $E: \mathcal{O}bj(\mathcal{S})\to\mathbb{R}^n$. More importantly, every morphism can be represented as a matrix $E: \mathcal{H}om(\mathcal{S})\to\mathbb{R}^{n}_{n}$. The encoding map $E$ is generally modeled by a $\textit{deep neural network}$. 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 $\textit{set-theoretic}$ approach. The goal of the current article is to promote the representation learning to a new level via a $\textit{category-theoretic}$ 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).