Separability and geometry of object manifolds in deep neural networks

Uri Cohen,Sueyeon Chung,Haim Sompolinsky,Daniel D Lee

doi:10.1038/s41467-020-14578-5

Abstract

Stimuli are represented in the brain by the collective population responses of sensory neurons, and an object presented under varying conditions gives rise to a collection of neural population responses called an ‘object manifold’. Changes in the object representation along a hierarchical sensory system are associated with changes in the geometry of those manifolds, and recent theoretical progress connects this geometry with ‘classification capacity’, a quantitative measure of the ability to support object classification. Deep neural networks trained on object classification tasks are a natural testbed for the applicability of this relation. We show how classification capacity improves along the hierarchies of deep neural networks with different architectures. We demonstrate that changes in the geometry of the associated object manifolds underlie this improved capacity, and shed light on the functional roles different levels in the hierarchy play to achieve it, through orchestrated reduction of manifolds’ radius, dimensionality and inter-manifold correlations.

Highlights

We consider here two types of manifolds: (1) “full class” manifolds, where all exemplars from the given class are used, or (2) “top 10%” manifolds, where just the 10% of the exemplars with large confidence in class-membership, as measured by the score achieved in the soft-max layer, at the node corresponding to the ground-truth class of the exemplar image in ImageNet
We consider here two types of manifolds: (1) “full class” manifolds, where all exemplars from the given class are used, or (2) “top 10%” manifolds, where just the 10% of the exemplars with large confidence in class-membership, as measured by the score achieved in the soft-max layer, at the node corresponding to the ground-truth class of the exemplar image in ImageNet (a pretrained AlexNet model from PyTorch implementation was used for the score throughout)
Networks with the same architectures but random weights do exhibit only slight improvement in capacity and manifold geometry, indicating that training rather than mere architecture is responsible for the successful reformatting of the manifolds

Summary

Objectives

The goal of our study was to delineate the contributions of computational, geometric, and correlation-based measures to the untangling of manifolds in deep networks, using a new theoretical framework

Methods

Results

Conclusion