Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Sharp Bounds for Multiple Models in Matrix Completion

Dali Liu, Haolei Weng

TL;DR

This work addresses matrix completion under a low-rank regime and shows that the long-standing $\log d$ gap between upper and minimax lower bounds can be eliminated. By leveraging sharp matrix concentration inequalities from Brailovskaya et al., the authors derive dimension-free, minimax-rate bounds for three common estimators across heavy-tailed, known-variance, and unknown-variance noise models. The results unify and sharpen prior bounds, demonstrating that the usual dimension-dependent factors are artifacts of spectral-norm analyses and not intrinsic to the problem. The technical contribution provides a path to sharper theoretical guarantees in high-dimensional matrix recovery, with potential applicability to broader sampling-with-replacement matrix problems and robust loss-based estimators.

Abstract

In this paper, we demonstrate how a class of advanced matrix concentration inequalities, introduced in \cite{brailovskaya2024universality}, can be used to eliminate the dimensional factor in the convergence rate of matrix completion. This dimensional factor represents a significant gap between the upper bound and the minimax lower bound, especially in high dimension. Through a more precise spectral norm analysis, we remove the dimensional factors for three popular matrix completion estimators, thereby establishing their minimax rate optimality.

Sharp Bounds for Multiple Models in Matrix Completion

TL;DR

This work addresses matrix completion under a low-rank regime and shows that the long-standing gap between upper and minimax lower bounds can be eliminated. By leveraging sharp matrix concentration inequalities from Brailovskaya et al., the authors derive dimension-free, minimax-rate bounds for three common estimators across heavy-tailed, known-variance, and unknown-variance noise models. The results unify and sharpen prior bounds, demonstrating that the usual dimension-dependent factors are artifacts of spectral-norm analyses and not intrinsic to the problem. The technical contribution provides a path to sharper theoretical guarantees in high-dimensional matrix recovery, with potential applicability to broader sampling-with-replacement matrix problems and robust loss-based estimators.

Abstract

In this paper, we demonstrate how a class of advanced matrix concentration inequalities, introduced in \cite{brailovskaya2024universality}, can be used to eliminate the dimensional factor in the convergence rate of matrix completion. This dimensional factor represents a significant gap between the upper bound and the minimax lower bound, especially in high dimension. Through a more precise spectral norm analysis, we remove the dimensional factors for three popular matrix completion estimators, thereby establishing their minimax rate optimality.

Paper Structure

This paper contains 13 sections, 16 theorems, 155 equations.

Table of Contents

  1. Introduction
  2. Main Results
  3. Matrix completion with heavy tailed noise
  4. Matrix completion with known variance
  5. Matrix completion with unknown variance of the noise
  6. Conclusion
  7. Proof
  8. Proof of Theorem \ref{['theorem: new heavy tail']}
  9. Proof of restricted strong convexity for Huber loss function
  10. Proof of Theorem \ref{['theorem: new lasso']}
  11. Proof of Theorem \ref{['theorem: new sq lasso']}
  12. Proof of sharp bounds for spectral norms
  13. Concentration of Maximum of Independent Random Variables

Key Result

Theorem 1

Let Assumptions assumption: mean var and assumption:X_i hold. We additionally assume $n\geq 160m\log^4 d$ and $\xi_i|X_i$ are symmetric. If we take with probability at least $1-\frac{2}{d}$, the following bound holds Here, $C_1$ and $C_2$ depend on the constants $L_2, L_3$ from Assumption assumption:X_i.

Theorems & Definitions (26)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4: Theorem 7 in klopp2014noisy
  • Theorem 5
  • Theorem 6: Theorem 10 in klopp2014noisy
  • Lemma 7
  • proof : Proof of Theorem \ref{['theorem: new heavy tail']}
  • Remark
  • Lemma 8
  • ...and 16 more