On the Symmetries of Deep Learning Models and their Internal Representations

TL;DR

This paper investigates how the symmetries inherent in neural network architectures influence the symmetries observed in their internal data representations, enhancing understanding of model behavior and interpretability.

Contribution

It introduces the concept of intertwiner groups to connect model architecture symmetries with internal data representations, supported by experimental analysis.

Findings

Symmetries of networks propagate into their data representations.

Intertwiner groups relate architecture symmetries to hidden state similarities.

Insights may justify focusing on activation bases for interpretability in ReLU networks.

Abstract

Symmetry is a fundamental tool in the exploration of a broad range of complex systems. In machine learning symmetry has been explored in both models and data. In this paper we seek to connect the symmetries arising from the architecture of a family of models with the symmetries of that family's internal representation of data. We do this by calculating a set of fundamental symmetry groups, which we call the intertwiner groups of the model. We connect intertwiner groups to a model's internal representations of data through a range of experiments that probe similarities between hidden states across models with the same architecture. Our work suggests that the symmetries of a network are propagated into the symmetries in that network's representation of data, providing us with a better understanding of how architecture affects the learning and prediction process. Finally, we speculate that for ReLU networks, the intertwiner groups may provide a justification for the common practice of concentrating model interpretability exploration on the activation basis in hidden layers rather than arbitrary linear combinations thereof.