On the Symmetries of Deep Learning Models and their Internal Representations

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authors	Charles Godfrey, Davis Brown, Tegan Emerson, Henry Kvnige
year	2022
url	https://arxiv.org/abs/2205.14258

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.

Notes:
- The following papers study the effect of weight space symmetries on training dynamics:
- Neural Mechanics - Symmetry and Broken Conservation Laws in Deep Learning Dynamics
- Understanding symmetries in deep networks
- G-SGD - Optimizing ReLU Neural Networks in its Positively Scale-Invariant Space
- Deep Learning Book