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On the Robustness of Cross-Concentrated Sampling for Matrix Completion

HanQin Cai, Longxiu Huang, Chandra Kundu, Bowen Su

TL;DR

This work tackles robust matrix completion under cross-concentrated sampling (CCS) by formulating Robust CCS Completion and introducing Robust CUR Completion (RCURC), a non-convex projected gradient descent method that leverages CUR decomposition to enforce a low-rank structure while isolating sparse outliers via decayed hard-thresholding. RCURC updates row- and column-derived components using observed residuals and uses a union-sum rule for the intersection to maintain a rank-$r$ approximation, with stopping based on a normalized residual error and stepsizes tied to observation rates. Empirical results on synthetic data and real video sequences demonstrate near-linear convergence and robust recovery under CCS with outliers, including PSNR values around 41.9 dB (shoppingmall) and 39.8 dB (restaurant). The paper provides a practical, scalable robust CCS framework and motivates future theoretical and application-driven work in sample complexity and broader domains.

Abstract

Matrix completion is one of the crucial tools in modern data science research. Recently, a novel sampling model for matrix completion coined cross-concentrated sampling (CCS) has caught much attention. However, the robustness of the CCS model against sparse outliers remains unclear in the existing studies. In this paper, we aim to answer this question by exploring a novel Robust CCS Completion problem. A highly efficient non-convex iterative algorithm, dubbed Robust CUR Completion (RCURC), is proposed. The empirical performance of the proposed algorithm, in terms of both efficiency and robustness, is verified in synthetic and real datasets.

On the Robustness of Cross-Concentrated Sampling for Matrix Completion

TL;DR

This work tackles robust matrix completion under cross-concentrated sampling (CCS) by formulating Robust CCS Completion and introducing Robust CUR Completion (RCURC), a non-convex projected gradient descent method that leverages CUR decomposition to enforce a low-rank structure while isolating sparse outliers via decayed hard-thresholding. RCURC updates row- and column-derived components using observed residuals and uses a union-sum rule for the intersection to maintain a rank- approximation, with stopping based on a normalized residual error and stepsizes tied to observation rates. Empirical results on synthetic data and real video sequences demonstrate near-linear convergence and robust recovery under CCS with outliers, including PSNR values around 41.9 dB (shoppingmall) and 39.8 dB (restaurant). The paper provides a practical, scalable robust CCS framework and motivates future theoretical and application-driven work in sample complexity and broader domains.

Abstract

Matrix completion is one of the crucial tools in modern data science research. Recently, a novel sampling model for matrix completion coined cross-concentrated sampling (CCS) has caught much attention. However, the robustness of the CCS model against sparse outliers remains unclear in the existing studies. In this paper, we aim to answer this question by exploring a novel Robust CCS Completion problem. A highly efficient non-convex iterative algorithm, dubbed Robust CUR Completion (RCURC), is proposed. The empirical performance of the proposed algorithm, in terms of both efficiency and robustness, is verified in synthetic and real datasets.
Paper Structure (11 sections, 2 theorems, 12 equations, 3 figures, 1 table, 2 algorithms)

This paper contains 11 sections, 2 theorems, 12 equations, 3 figures, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. Related Work
  3. Notation
  4. Preliminaries
  5. Assumptions
  6. CUR Approximation
  7. Proposed Algorithm
  8. Numerical Experiments
  9. Synthetic Datasets
  10. Video Background Subtraction
  11. Conclusion Remarks

Key Result

Theorem 3

Consider row and column index sets ${\mathcal{I}},{\mathcal{J}}\subseteq[n]$. Denote submatrices ${\bm{C}}=[\bm{X}]_{:,{\mathcal{J}}}$, ${\bm{U}}=[\bm{X}]_{{\mathcal{I}},{\mathcal{J}}}$ and ${\bm{R}}=[\bm{X}]_{{\mathcal{I}},:}$. If $\mathrm{rank}({\bm{U}})=\mathrm{rank}(\bm{X})$, then $\bm{X} = {\bm

Figures (3)

  • Figure 1: cai2023ccs Visual illustrations of different sampling schemes. From left to right, sampling methods change from the uniform sampling style to the CUR sampling style with the same total observation rate. Colored pixels indicate observed entries, while black pixels mean missing entries.
  • Figure 2: Empirical convergence of RCURC. Left:$c=10$ and varying $r$. Right:$r=5$ and varying $c$.
  • Figure 3: Video background subtraction results: The first row shows shoppingmall, the second restaurant. Column one displays the ground truth background via IRCUR from full datasets. Columns two and three show two observed frames from our CCS model. The last column presents the background from RCURC using partial videos.

Theorems & Definitions (2)

  • Theorem 3: HH2020
  • Theorem 4: chiu2013sublinear