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Robust Matrix Completion with Deterministic Sampling via Convex Optimization

Yinjian Wang

TL;DR

This paper proposes \textit{restricted approximate $\infty$-isometry property} and proves that, if a low-rank matrix and a sparse matrix from the compressed counterpart of their superposition satisfy such property, the exact recovery from its sampled counterpart grossly corrupted by a small fraction of outliers via convex optimization happens with very high probability.

Abstract

This paper deals with the problem of robust matrix completion -- retrieving a low-rank matrix and a sparse matrix from the compressed counterpart of their superposition. Though seemingly not an unresolved issue, we point out that the compressed matrix in our case is sampled in a deterministic pattern instead of those random ones on which existing studies depend. In fact, deterministic sampling is much more hardware-friendly than random ones. The limited resources on many platforms leave deterministic sampling the only choice to sense a matrix, resulting in the significance of investigating robust matrix completion with deterministic pattern. In such spirit, this paper proposes \textit{restricted approximate $\infty$-isometry property} and proves that, if a \textit{low-rank} and \textit{incoherent} square matrix and certain deterministic sampling pattern satisfy such property and two existing conditions called \textit{isomerism} and \textit{relative well-conditionedness}, the exact recovery from its sampled counterpart grossly corrupted by a small fraction of outliers via convex optimization happens with very high probability.

Robust Matrix Completion with Deterministic Sampling via Convex Optimization

TL;DR

This paper proposes \textit{restricted approximate -isometry property} and proves that, if a low-rank matrix and a sparse matrix from the compressed counterpart of their superposition satisfy such property, the exact recovery from its sampled counterpart grossly corrupted by a small fraction of outliers via convex optimization happens with very high probability.

Abstract

This paper deals with the problem of robust matrix completion -- retrieving a low-rank matrix and a sparse matrix from the compressed counterpart of their superposition. Though seemingly not an unresolved issue, we point out that the compressed matrix in our case is sampled in a deterministic pattern instead of those random ones on which existing studies depend. In fact, deterministic sampling is much more hardware-friendly than random ones. The limited resources on many platforms leave deterministic sampling the only choice to sense a matrix, resulting in the significance of investigating robust matrix completion with deterministic pattern. In such spirit, this paper proposes \textit{restricted approximate -isometry property} and proves that, if a \textit{low-rank} and \textit{incoherent} square matrix and certain deterministic sampling pattern satisfy such property and two existing conditions called \textit{isomerism} and \textit{relative well-conditionedness}, the exact recovery from its sampled counterpart grossly corrupted by a small fraction of outliers via convex optimization happens with very high probability.
Paper Structure (14 sections, 8 theorems, 51 equations)

This paper contains 14 sections, 8 theorems, 51 equations.

Table of Contents

  1. Introduction
  2. Notations
  3. Problem Formulation
  4. Exact Identifiability Theories
  5. Restricted Approximate $\infty$-Isometry Property
  6. Main Theorem
  7. Proof Architecture of the Main Theorem
  8. Mathematical Proofs
  9. Preparation Theories
  10. Proofs of Lemma \ref{['Le: Relaxed KKT']}-\ref{['Le: Certificate2']}
  11. Proof of Lemma \ref{['Le: Relaxed KKT']}
  12. Proof of Lemma \ref{['Le: Certificate1']}
  13. Proof of Lemma \ref{['Le: Certificate2']}
  14. Conclusions

Key Result

Theorem 4.1

With the premises in Section sec:formulation, if $\mathcal{P}_{\mathcal{O}}$ satisfies RAIIP, additionally if $\rho<\min\left\{\rho_s,1-C\frac{\nu r\log^2(n)}{n}\right\}$ with $C$ some positive numerical constant, then there exists another constant $C_m>0$, such that $\left(\boldsymbol{L}_0,\overlin

Theorems & Definitions (20)

  • Definition 4.1: Restricted $\infty$-norm of linear operators
  • Definition 4.2: Restricted Approximate $\infty$-Isometry Property
  • Theorem 4.1
  • Lemma 4.1
  • proof
  • Lemma 4.2
  • proof
  • Lemma 4.3
  • proof
  • Proposition 5.1
  • ...and 10 more