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Free denoising via overlap measures and c-freeness techniques

Maxime Fevrier, Alexandru Nica, Kamil Szpojankowski

TL;DR

This work develops a noncommutative analogue of conditional expectation (free denoising) by introducing overlap measures for pairs of selfadjoint variables and deriving explicit formulas for $E(a\mid P(a,b))$ in additive and multiplicative free settings. Central to the approach are subordination functions and a disintegration of the overlap measure, which yield Radon–Nikodym derivatives that serve as denoisers. The paper connects these results to conditional freeness (c-freeness) and shows how general noise $P(a,b)$ reduces to a two-state framework, with concrete instances including the free Tweedie formulas and the Ledoit–Péché shrinkage estimators in matrix denoising. It also outlines a bridge to matrix denoising, including asymptotic optimality results and applications to covariance estimation. Overall, the overlap-measure methodology unifies additive/mmultiplicative denoising and links free probability techniques with practical matrix-denoising insights.

Abstract

We study the problem of free denoising. For free selfadjoint random variables $a,b$, where we interpret $a$ as a signal and $b$ as noise, we find $E(a|a+b)$. To that end, we study a probability measure $μ^{( \mathrm{ov} )}_{a,a+b}$ on $\mathbb{R}^2$ which we call the overlap measure. We show that $μ^{( \mathrm{ov} )}_{a,a+b}$ is absolutely continuous with respect to the product measure $μ_a\times μ_{a+b}$. The Radon-Nikodym derivative gives direct access to $E(a|a+b)$. We show that analogous results hold in the case of multiplicative noise when $a,b$ are positive and the aim is to find $E(a|a^{1/2}ba^{1/2})$. In a parallel development we show that, for a general selfadjoint expression $P(a,b)$ made with $a$ and $b$, finding $E(a|P(a,b))$ is equivalent to finding the distribution of $P(a,b)$ in a certain two-state probability space $(\mathcal{A},\varphi,χ)$, where $a,b$ are c-free with respect to $(\varphi,χ)$ in the sense of Bożejko-Leinert-Speicher. We discuss how free denoising (which is set in the framework of an abstract $W^{*}$-probability space) relates to the notion of ''matrix denoising'' previously discussed in the random matrix literature.

Free denoising via overlap measures and c-freeness techniques

TL;DR

This work develops a noncommutative analogue of conditional expectation (free denoising) by introducing overlap measures for pairs of selfadjoint variables and deriving explicit formulas for in additive and multiplicative free settings. Central to the approach are subordination functions and a disintegration of the overlap measure, which yield Radon–Nikodym derivatives that serve as denoisers. The paper connects these results to conditional freeness (c-freeness) and shows how general noise reduces to a two-state framework, with concrete instances including the free Tweedie formulas and the Ledoit–Péché shrinkage estimators in matrix denoising. It also outlines a bridge to matrix denoising, including asymptotic optimality results and applications to covariance estimation. Overall, the overlap-measure methodology unifies additive/mmultiplicative denoising and links free probability techniques with practical matrix-denoising insights.

Abstract

We study the problem of free denoising. For free selfadjoint random variables , where we interpret as a signal and as noise, we find . To that end, we study a probability measure on which we call the overlap measure. We show that is absolutely continuous with respect to the product measure . The Radon-Nikodym derivative gives direct access to . We show that analogous results hold in the case of multiplicative noise when are positive and the aim is to find . In a parallel development we show that, for a general selfadjoint expression made with and , finding is equivalent to finding the distribution of in a certain two-state probability space , where are c-free with respect to in the sense of Bożejko-Leinert-Speicher. We discuss how free denoising (which is set in the framework of an abstract -probability space) relates to the notion of ''matrix denoising'' previously discussed in the random matrix literature.
Paper Structure (29 sections, 26 theorems, 166 equations)

This paper contains 29 sections, 26 theorems, 166 equations.

Table of Contents

  1. Introduction
  2. Problem description.
  3. Motivation: analogies with classical probability.
  4. Relation to subordination functions.
  5. Overlap measure and overlap function.
  6. Description of results -- additive case.
  7. Description of results -- multiplicative case.
  8. General noise and conditional freeness.
  9. Relation to matrix denoising.
  10. Organization of the paper.
  11. The framework of a Borel-ncps
  12. Overlap measure and overlap function
  13. Conditional expectations, via disintegration of the overlap measure
  14. Review of some subordination results in free probability
  15. Review of subordination results -- additive case.
  16. ...and 14 more sections

Key Result

Theorem 1.1

Let $a,b$ be freely independent selfadjoint elements in a $W^{*}$-pro-ba-bi-li-ty space $( \mathcal{A} , \varphi )$. Let $\mu$ and respectively $\nu$ be the distributions of $a$ and $b$ (with respect to $\varphi$), and assume that neither of $\mu, \nu$ is a point mass. Note that, due to the free in -- For $t \in U$ one has

Theorems & Definitions (62)

  • Theorem 1.1
  • Theorem 1.2
  • Proposition 1.3
  • Remark 1.4
  • Theorem 1.5
  • Example 1.6
  • Theorem 1.7
  • Definition 2.3
  • Remark 2.4
  • Example 2.7
  • ...and 52 more