Metric Space Magnitude and Generalisation in Neural Networks
This work addresses the challenge of interpreting neural networks for researchers and practitioners, offering a novel theoretical framework to assess generalization, though it appears incremental in applying an existing mathematical concept to a new context.
The paper tackles the problem of understanding and quantifying the learning process in deep neural networks by introducing magnitude, a topological invariant, to study internal representations and predict generalization capabilities, demonstrating experimentally that it serves as a good indicator of generalization error.
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.