Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Entry-Specific Bounds for Low-Rank Matrix Completion under Highly Non-Uniform Sampling

Xumei Xi, Christina Lee Yu, Yudong Chen

TL;DR

This work addresses entrywise uncertainty in low-rank matrix completion under highly non-uniform sampling. It introduces a submatrix completion meta-algorithm that for each target entry selects an appropriate submatrix and runs a matrix-estimation method (e.g., SVT), yielding refined, entrywise error rates. The authors derive an entrywise upper bound that adapts to localized sampling probabilities and prove a matching minimax lower bound under structured sampling, demonstrating near-optimality in many regimes. Numerical experiments on block-structured and rank-one sampling patterns confirm substantial empirical gains over applying SVT to the full matrix, highlighting the practical impact of localized, per-entry analysis in non-uniform settings.

Abstract

Low-rank matrix completion concerns the problem of estimating unobserved entries in a matrix using a sparse set of observed entries. We consider the non-uniform setting where the observed entries are sampled with highly varying probabilities, potentially with different asymptotic scalings. We show that under structured sampling probabilities, it is often better and sometimes optimal to run estimation algorithms on a smaller submatrix rather than the entire matrix. In particular, we prove error upper bounds customized to each entry, which match the minimax lower bounds under certain conditions. Our bounds characterize the hardness of estimating each entry as a function of the localized sampling probabilities. We provide numerical experiments that confirm our theoretical findings.

Entry-Specific Bounds for Low-Rank Matrix Completion under Highly Non-Uniform Sampling

TL;DR

This work addresses entrywise uncertainty in low-rank matrix completion under highly non-uniform sampling. It introduces a submatrix completion meta-algorithm that for each target entry selects an appropriate submatrix and runs a matrix-estimation method (e.g., SVT), yielding refined, entrywise error rates. The authors derive an entrywise upper bound that adapts to localized sampling probabilities and prove a matching minimax lower bound under structured sampling, demonstrating near-optimality in many regimes. Numerical experiments on block-structured and rank-one sampling patterns confirm substantial empirical gains over applying SVT to the full matrix, highlighting the practical impact of localized, per-entry analysis in non-uniform settings.

Abstract

Low-rank matrix completion concerns the problem of estimating unobserved entries in a matrix using a sparse set of observed entries. We consider the non-uniform setting where the observed entries are sampled with highly varying probabilities, potentially with different asymptotic scalings. We show that under structured sampling probabilities, it is often better and sometimes optimal to run estimation algorithms on a smaller submatrix rather than the entire matrix. In particular, we prove error upper bounds customized to each entry, which match the minimax lower bounds under certain conditions. Our bounds characterize the hardness of estimating each entry as a function of the localized sampling probabilities. We provide numerical experiments that confirm our theoretical findings.
Paper Structure (28 sections, 10 theorems, 52 equations, 4 figures)

This paper contains 28 sections, 10 theorems, 52 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Related Work
  3. Problem setup
  4. Latent Variable Model
  5. Monotone Sampling Probabilities
  6. Algorithm: submatrix completion
  7. Illustrating example
  8. Theoretical Guarantees
  9. Upper bound for our algorithm
  10. Minimax lower bound
  11. Example: block-structured $P$
  12. Key ideas of proof
  13. Guarantee for SVT with non-uniform observations
  14. Improved bound under a few small probabilities
  15. Numerical experiments
  16. ...and 13 more sections

Key Result

Lemma 1

Under Assumption assum:mono, if $i \le i^\ast$ and $j \le i^\ast$, then $k^\ast (i,j) = i^\ast$.

Figures (4)

  • Figure 1: An example of using submatrix completion with monotone $P$.
  • Figure 2: Heatmaps for Subsection \ref{['subsec:block-constant']}.
  • Figure 3: Heatmaps for Subsection \ref{['subsec:rank-1']}.
  • Figure 4: Histograms of relative improvement.

Theorems & Definitions (10)

  • Lemma 1
  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Lemma 2: Low-rank signal
  • Lemma 3: Bounded signal
  • Lemma 4: Incoherence & condition number guarantee
  • Lemma 5
  • Lemma 6