A Case Study of Low Ranked Self-Expressive Structures in Neural Network Representations

Uday Singh Saini, William Shiao, Yahya Sattar, Yogesh Dahiya, Samet Oymak, Evangelos E. Papalexakis

Keywords: Subspace Clustering, Centered Kernel Alignment, Representation Similarity Measures.

TL;DR: The paper establishes connections between subspace clustering based and linear kernel based similarity measures to analyse neural representations and demonstrates their usefulness via various experiments.

Abstract: Understanding neural networks by studying their underlying geometry can help us understand their embedded inductive priors and representation capacity. Prior representation analysis tools like (Linear) Centered Kernel Alignment (CKA) offer a lens to probe those structures via a kernel similarity framework. In this work we approach the problem of understanding the underlying geometry via the lens of subspace clustering, where each input is represented as a linear combination of other inputs. Such structures are called self-expressive structures. In this work we analyze their evolution and gauge their usefulness with the help of linear probes. We also demonstrate a close relationship between subspace clustering and linear CKA and demonstrate its utility to act as a more sensitive similarity measure of representations when compared with linear CKA. We do so by comparing the sensitivities of both measures to changes in representation across their singular value spectrum, by analyzing the evolution of self-expressive structures in networks trained to generalize and memorize and via a comparison of networks trained with different optimization objectives. This analysis helps us ground the utility of subspace clustering based approaches to analyze neural representations and motivate future work on exploring the utility of enforcing similarity between self-expressive structures as a means of training neural networks.

Submission Number: 52