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

arXiv: https://arxiv.org/abs/1807.00349 · HTML: https://mlsanity.com/paper/1807.00349/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 1 Jul 2018.

## Built on by (most cited first, 5 of 5)

- [Machine Learning in LiDAR 3D point clouds](https://mlsanity.com/paper/2101.09318/) (2021)
- [Integrating Product Coefficients for Improved 3D LiDAR Data Classification](https://mlsanity.com/paper/2503.11943/) (2025)
- [Integrating Product Coefficients for Improved 3D LiDAR Data Classification (Part II)](https://mlsanity.com/paper/2510.15219/) (2025)
- [Data-induced multiscale losses and efficient multirate gradient descent schemes](https://mlsanity.com/paper/2402.03021/) (2024)
- [Neighbor Embeddings Using Unbalanced Optimal Transport Metrics](https://mlsanity.com/paper/2509.19226/) (2025)

## Abstract (arXiv)

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.