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Spectral gaps and measure decompositions

March T. Boedihardjo, Joe Kileel, Vandy Tombs

Abstract

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

Spectral gaps and measure decompositions

Abstract

Let be a probability measure on . In this paper, we introduce a new set of computable quantities in that are invariant under orthogonal transformations, namely, the eigenvalues of the 4th moment operator of . We show how the first and second largest eigenvalues of this operator can determine the extent to which can be decomposed as an equal weight mixture of two probability measures with significantly different second order statistics.
Paper Structure (13 sections, 18 theorems, 135 equations)

This paper contains 13 sections, 18 theorems, 135 equations.

Table of Contents

  1. Introduction
  2. Overview
  3. Geometric vs statistical objectives
  4. Examples of the spectral gap phenomenon
  5. The $L^{p}$-$L^{2}$ equivalence assumption
  6. First main result: largest eigenvalue
  7. Second main result
  8. Unequal weight mixtures
  9. Examples
  10. Proof of the first main result
  11. Proof of the upper bound in the second main result
  12. Proof of the lower bound in the second main result
  13. Unequal weight mixtures

Key Result

Theorem 1.12

Let $\mu\in\mathcal{P}_{4}(\mathbb{R}^{d})$ and $B=\int_{\mathbb{R}^{d}}xx^{T}\,d\mu(x)$. Then Moreover, if $\beta\geq 1$ satisfies for all $v\in\mathbb{R}^{d}$, then

Theorems & Definitions (61)

  • Definition 1.1
  • Remark 1.2
  • Example 1.3
  • Example 1.4
  • Example 1.5
  • Example 1.6
  • Example 1.7
  • Example 1.8
  • Example 1.9
  • Example 1.10
  • ...and 51 more