Spectral gaps and measure decompositions

TL;DR

This paper introduces eigenvalues of a 4th moment operator as invariant measures to analyze the decomposability of probability measures in , providing new tools for understanding measure structure.

Contribution

It proposes a novel set of computable, orthogonally invariant quantities based on the 4th moment operator to study measure decompositions.

Findings

Eigenvalues determine the extent of measure decomposition.

First and second eigenvalues reveal differences in second order statistics.

New invariants facilitate analysis of measure structure.

Abstract

Let $\mu$ be a probability measure on $\mathbb{R}^{d}$. In this paper, we introduce a new set of computable quantities in $\mu$ that are invariant under orthogonal transformations, namely, the eigenvalues of the 4th moment operator of $\mu$. We show how the first and second largest eigenvalues of this operator can determine the extent to which $\mu$ can be decomposed as an equal weight mixture of two probability measures with significantly different second order statistics.