Persistent homology for high-dimensional data based on spectral methods

Sebastian Damrich, Philipp Berens, Dmitry Kobak

Primary Area: general machine learning (i.e., none of the above)

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Keywords: persistent homology, topological data analysis, diffusion distance, effective resistance, single-cell RNA sequencing, curse of dimensionality, spectral methods, topology

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TL;DR: Vanilla persistent homology and its standard extensions fail on high-dimensional data, but persistent homology with spectral distances works well.

Abstract: Persistent homology is a popular computational tool for detecting non-trivial topology of point clouds, such as the presence of loops or voids. However, many real-world datasets with low intrinsic dimensionality reside in an ambient space of much higher dimensionality. We show that in this case vanilla persistent homology becomes very sensitive to noise and fails to detect the correct topology. The same holds true for most existing refinements of persistent homology. As a remedy, we find that spectral distances on the $k$-nearest-neighbor graph of the data, such as diffusion distance and effective resistance, allow persistent homology to detect the correct topology even in the presence of high-dimensional noise. Furthermore, we derive a novel closed-form expression for effective resistance in terms of the eigendecomposition of the graph Laplacian, and describe its relation to diffusion distances. Finally, we apply these methods to several high-dimensional single-cell RNA-sequencing datasets and show that spectral distances on the $k$-nearest-neighbor graph allow robust detection of cell cycle loops.

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Submission Number: 3784