Metric Space Magnitude and Generalisation in Neural Networks

Rayna Andreeva, Katharina Limbeck, Bastian Rieck, Rik Sarkar

Keywords: magnitude, metric spaces, neural networks, generalisation

TL;DR: We propose the usage of a novel invariant, called magnitude, to quantify the learning process of deep neural networks, and we demonstrate that it works both theoretically and empirically.

Abstract: Deep learning models have seen significant successes in numerous applications, but their inner workings remain elusive. The purpose of this work is to quantify the learning process of deep neural networks through the lens of a novel topological invariant called magnitude. Magnitude is an isometry invariant; its properties are an active area of research as it encodes many known invariants of a metric space. We use magnitude to study the internal representations of neural networks and propose a new method for determining their generalisation capabilities. Moreover, we theoretically connect magnitude dimension and the generalisation error, and demonstrate experimentally that the proposed framework can be a good indicator of the latter.

Type Of Submission: Proceedings Track (8 pages)

Submission Number: 29