A Group Theoretic Perspective on Unsupervised Deep Learning

TL;DR

This paper introduces a novel group-theoretic framework to understand how deep learning models learn simple features first and develop complex, higher-order representations through layer-wise pretraining, explaining the emergence of hierarchical features.

Contribution

It establishes a connection between pretraining in deep learning and group theory, particularly orbits and stabilizers, proposing shadow groups to analyze feature simplicity and hierarchy.

Findings

Pretraining aligns with searching for features with minimal group orbits.

Shadow groups approximate neural network transformations, revealing feature simplicity.

Deeper layers capture higher-order, more complex representations.

Abstract

Why does Deep Learning work? What representations does it capture? How do higher-order representations emerge? We study these questions from the perspective of group theory, thereby opening a new approach towards a theory of Deep learning. One factor behind the recent resurgence of the subject is a key algorithmic step called {\em pretraining}: first search for a good generative model for the input samples, and repeat the process one layer at a time. We show deeper implications of this simple principle, by establishing a connection with the interplay of orbits and stabilizers of group actions. Although the neural networks themselves may not form groups, we show the existence of {\em shadow} groups whose elements serve as close approximations. Over the shadow groups, the pre-training step, originally introduced as a mechanism to better initialize a network, becomes equivalent to a search for features with minimal orbits. Intuitively, these features are in a way the {\em simplest}. Which explains why a deep learning network learns simple features first. Next, we show how the same principle, when repeated in the deeper layers, can capture higher order representations, and why representation complexity increases as the layers get deeper.