Neural Networks as Paths through the Space of Representations

TL;DR

This paper introduces a geometric framework to interpret deep neural networks as paths through a high-dimensional representation space, using metric representational similarity to analyze and visualize model behaviors.

Contribution

It formalizes the concept of neural network layer transformations as paths in representational space using advanced geometric measures, enabling new insights into model interpretability.

Findings

Visualized paths of ResNet and VGG on CIFAR-10

Extended representational similarity with geodesics and angles

Proposed geometric tools for understanding training and model similarities

Abstract

Deep neural networks implement a sequence of layer-by-layer operations that are each relatively easy to understand, but the resulting overall computation is generally difficult to understand. We consider a simple hypothesis for interpreting the layer-by-layer construction of useful representations: perhaps the role of each layer is to reformat information to reduce the "distance" to the desired outputs. With this framework, the layer-wise computation implemented by a deep neural network can be viewed as a path through a high-dimensional representation space. We formalize this intuitive idea of a "path" by leveraging recent advances in *metric* representational similarity. We extend existing representational distance methods by computing geodesics, angles, and projections of representations, going beyond mere layer distances. We then demonstrate these tools by visualizing and comparing the paths taken by ResNet and VGG architectures on CIFAR-10. We conclude by sketching additional ways that this kind of representational geometry can be used to understand and interpret network training, and to describe novel kinds of similarities between different models.