Metric Space Magnitude for Evaluating Unsupervised Representation Learning

Katharina Limbeck, Rayna Andreeva, Rik Sarkar, Bastian Rieck

Primary Area: unsupervised, self-supervised, semi-supervised, and supervised representation learning

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Keywords: magnitude, metric spaces, representation learning, geometry

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TL;DR: We use the magnitude of a metric space, a recently-developed invariant, to efficiently analyse embeddings of high-dimensional data sets.

Abstract:

The magnitude of a metric space was recently established as
a novel invariant, providing a measure of the `effective size' of
a space across multiple scales.
By capturing both geometrical and topological properties of
data, magnitude is poised to address challenges in unsupervised
representation learning tasks.
We formalise a novel notion of a dissimilarity between magnitude
functions of finite metric spaces and use them to derive a quality
measure for dimensionality reduction tasks.
Our measure is provably stable under perturbations of the data, can be
efficiently calculated, and enables a rigorous multi-scale comparison of
embeddings.
We show the utility of our measure in an experimental suite that
comprises different domains and tasks, including the comparison of
data visualisations.

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Submission Number: 5963