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The Masked Matrix Separation Problem: A First Analysis

Xuemei Chen, Rongrong Wang

TL;DR

This work extends robust PCA to a masked setting where the observed matrix is $M_0 = L_0 + H S_0$ with a known mask $H$. It proposes a convex program minimizing $\gamma\|S\|_1 + \|L\|_*$ subject to $L + H S = M_0$ and develops a theory around a restricted infinity norm property (RINP) to guarantee exact recovery under separability and dual-certificate conditions. The paper derives deterministic guarantees under a scaled-$S$-$\delta$-RINP and incoherence bounds, and analyzes special mask regimes (orthogonal columns, unit condition, Gaussian masks) with concrete bounds. Numerical experiments on blurring masks, random masks, and simulated electrodermal activity (EDA) decomposition validate the approach and illustrate how sparsity and rank impact recoverability under masking.

Abstract

Given a known matrix that is the sum of a low rank matrix and a masked sparse matrix, we wish to recover both the low rank component and the sparse component. The sparse matrix is masked in the sense that a linear transformation has been applied on its left. We propose a convex optimization problem to recover the low rank and sparse matrices, which generalizes the robust PCA framework. We provide incoherence conditions for the success of the proposed convex optimizaiton problem, adapting to the masked setting. The ``mask'' matrix can be quite general as long as a so-called restricted infinity norm condition is satisfied. Further analysis on the incoherence condition is provided and we conclude with promising numerical experiments.

The Masked Matrix Separation Problem: A First Analysis

TL;DR

This work extends robust PCA to a masked setting where the observed matrix is with a known mask . It proposes a convex program minimizing subject to and develops a theory around a restricted infinity norm property (RINP) to guarantee exact recovery under separability and dual-certificate conditions. The paper derives deterministic guarantees under a scaled---RINP and incoherence bounds, and analyzes special mask regimes (orthogonal columns, unit condition, Gaussian masks) with concrete bounds. Numerical experiments on blurring masks, random masks, and simulated electrodermal activity (EDA) decomposition validate the approach and illustrate how sparsity and rank impact recoverability under masking.

Abstract

Given a known matrix that is the sum of a low rank matrix and a masked sparse matrix, we wish to recover both the low rank component and the sparse component. The sparse matrix is masked in the sense that a linear transformation has been applied on its left. We propose a convex optimization problem to recover the low rank and sparse matrices, which generalizes the robust PCA framework. We provide incoherence conditions for the success of the proposed convex optimizaiton problem, adapting to the masked setting. The ``mask'' matrix can be quite general as long as a so-called restricted infinity norm condition is satisfied. Further analysis on the incoherence condition is provided and we conclude with promising numerical experiments.
Paper Structure (19 sections, 12 theorems, 94 equations, 3 figures)

This paper contains 19 sections, 12 theorems, 94 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Applications
  3. Related Work and Contributions
  4. Main Results
  5. Notations and Preliminaries
  6. Separable Conditions
  7. Restricted Infinity Norm Property
  8. When columns of $H$ are orthogonal
  9. When $\|H\|\|H^\dagger\|=1$
  10. $H$ is Gaussian
  11. Entrywise bound.
  12. Action on sparse vectors.
  13. Conditions on low rank and sparse matrices
  14. Incoherence condition
  15. Random sparse $S$ and low rank $L$ with a Gaussian $H$
  16. ...and 4 more sections

Key Result

Proposition 2.3

Given any $L_0\in\mathbb{R}^{m\times n}, S_0\in\mathbb{R}^{p\times n}$, and let $H$ satisfy the scaled $S_0$-$\delta$-RINP equ:Grip. If $\mu_G(S_0)\xi_G(L_0)<1-\delta$, then $H\Omega(S_0)\cap \mathcal{T}(L_0)=\{0\}$.

Figures (3)

  • Figure 1: Relative errors when $H$ is a blurring mask
  • Figure 2: Relative errors when $H$ is random
  • Figure 3: Relative error in the EDA experiment as sparsity varies

Theorems & Definitions (26)

  • Definition 2.1
  • Definition 2.2
  • Proposition 2.3
  • proof
  • Proposition 2.4
  • proof
  • Corollary 2.5
  • proof
  • Lemma 2.6
  • proof
  • ...and 16 more