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.

翻译：深度学习模型在众多应用中取得了显著成功，但其内部工作机制仍难以捉摸。本文旨在通过一种名为“幅度”的新型拓扑不变量来量化深度神经网络的学习过程。幅度是一种等距不变量；由于它编码了度量空间的许多已知不变量，其性质是当前研究的热点。我们利用幅度来研究神经网络的内部表征，并提出一种确定其泛化能力的新方法。此外，我们从理论上建立了幅度维度与泛化误差之间的联系，并通过实验证明，所提出的框架能够很好地指示后者。