A Group Theoretic Perspective on Unsupervised Deep Learning
This work provides a foundational theoretical framework for deep learning, potentially impacting all of ML/AI by offering new insights into representation learning.
The authors tackled the problem of understanding why deep learning works and what representations it captures by applying group theory to analyze unsupervised pretraining, showing that pretraining corresponds to searching for features with minimal orbits, which explains why simple features are learned first and how higher-order representations emerge in deeper layers.
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.