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

TL;DR

This paper introduces a heuristic, multiscale testing framework for the multi-manifold hypothesis, enabling better understanding of data's intrinsic structure through variance-based dimension analysis and spline interpolation, demonstrated on real data.

Contribution

It presents a novel multiscale hypothesis testing framework for multi-manifold structures, integrating spline interpolation and variance-based intrinsic dimension estimation.

Findings

Effective in identifying multi-manifold structures in empirical data

Applicable to real-world data analysis scenarios

Enhances understanding of data's intrinsic geometry

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.