Making Laplacians commute

TL;DR

This paper introduces a method to construct commuting Laplacian operators for multimodal spectral geometry, enabling better analysis of multi-modal data through shared eigenbases, with applications in dimensionality reduction, shape analysis, and clustering.

Contribution

We develop a novel approach to find closest commuting operators to given Laplacians, extending spectral analysis tools for multimodal data analysis.

Findings

Improved capturing of multi-modal data structure

Enhanced performance in spectral clustering and shape analysis

Validated on synthetic and real datasets

Abstract

In this paper, we construct multimodal spectral geometry by finding a pair of closest commuting operators (CCO) to a given pair of Laplacians. The CCOs are jointly diagonalizable and hence have the same eigenbasis. Our construction naturally extends classical data analysis tools based on spectral geometry, such as diffusion maps and spectral clustering. We provide several synthetic and real examples of applications in dimensionality reduction, shape analysis, and clustering, demonstrating that our method better captures the inherent structure of multi-modal data.