Statistical analysis of random objects via metric measure Laplacians

TL;DR

This paper introduces a spectral analysis framework for complex metric measure spaces using a convolutional Laplacian, enabling principled comparison and feature extraction of objects with applications demonstrated through examples.

Contribution

It develops a novel spectral analysis method for metric measure spaces, defining mean spectral measures and exploring their statistical properties.

Findings

Spectral signatures effectively characterize complex objects.

The proposed methods facilitate object comparison and feature extraction.

Statistical properties of spectral measures are rigorously derived.

Abstract

In this paper, we consider a certain convolutional Laplacian for metric measure spaces and investigate its potential for the statistical analysis of complex objects. The spectrum of that Laplacian serves as a signature of the space under consideration and the eigenvectors provide the principal directions of the shape, its harmonics. These concepts are used to assess the similarity of objects or understand their most important features in a principled way which is illustrated in various examples. Adopting a statistical point of view, we define a mean spectral measure and its empirical counterpart. The corresponding limiting process of interest is derived and statistical applications are discussed.