Title: Heuristic Framework for Multi-Scale Testing of the Multi-Manifold Hypothesis,
Abstract: When analyzing empirical data, we often find that global linear models
overestimate the number of parameters required. In such cases, we may ask
whether the data lies on or near a manifold or a set of manifolds (a so-called
multi-manifold) of lower dimension than the ambient space. This question can be
phrased as a (multi-) manifold hypothesis. The identification of such intrinsic
multiscale features is a cornerstone of data analysis and representation and
has given rise to a large body of work on manifold learning. In this work, we
review key results on multi-scale data analysis and intrinsic dimension
followed by the introduction of a heuristic, multiscale framework for testing
the multi-manifold hypothesis. Our method implements a hypothesis test on a set
of spline-interpolated manifolds constructed from variance-based intrinsic
dimensions. The workflow is suitable for empirical data analysis as we
demonstrate on two use cases.