Heuristic Framework for Multi-Scale Testing of the Multi-Manifold Hypothesis
  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.
