  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.
