Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Adversarial Robust Low Rank Matrix Estimation: Compressed Sensing and Matrix Completion

Takeyuki Sasai, Hironori Fujisawa

TL;DR

This work develops robust low-rank matrix estimation under adversarial contamination within trace regression, unifying matrix compressed sensing and matrix completion through a single estimator that combines the Huber loss with nuclear-norm regularization. By introducing matrix-restricted eigenvalue (MRE) conditions and accommodating heavy-tailed noise via $L$-subGaussian and subWeibull models, the authors obtain sharp high-probability error bounds that improve upon prior results and extend robustness to broader noise regimes. The main contributions include a deterministic convex-analytic framework and three main results: (i) robust matrix CS with sharp Frobenius-bound guarantees, (ii) adversarial lasso with separated stochastic and sparsity terms, and (iii) robust matrix completion under two noise models with spikiness constraints. Overall, the paper provides practical, theoretically grounded tools for robust recovery of low-rank structures in contaminated data, with implications for compressed sensing, collaborative filtering, and related matrix-analytic tasks.

Abstract

We consider robust low rank matrix estimation as a trace regression when outputs are contaminated by adversaries. The adversaries are allowed to add arbitrary values to arbitrary outputs. Such values can depend on any samples. We deal with matrix compressed sensing, including lasso as a partial problem, and matrix completion, and then we obtain sharp estimation error bounds. To obtain the error bounds for different models such as matrix compressed sensing and matrix completion, we propose a simple unified approach based on a combination of the Huber loss function and the nuclear norm penalization, which is a different approach from the conventional ones. Some error bounds obtained in the present paper are sharper than the past ones.

Adversarial Robust Low Rank Matrix Estimation: Compressed Sensing and Matrix Completion

TL;DR

This work develops robust low-rank matrix estimation under adversarial contamination within trace regression, unifying matrix compressed sensing and matrix completion through a single estimator that combines the Huber loss with nuclear-norm regularization. By introducing matrix-restricted eigenvalue (MRE) conditions and accommodating heavy-tailed noise via -subGaussian and subWeibull models, the authors obtain sharp high-probability error bounds that improve upon prior results and extend robustness to broader noise regimes. The main contributions include a deterministic convex-analytic framework and three main results: (i) robust matrix CS with sharp Frobenius-bound guarantees, (ii) adversarial lasso with separated stochastic and sparsity terms, and (iii) robust matrix completion under two noise models with spikiness constraints. Overall, the paper provides practical, theoretically grounded tools for robust recovery of low-rank structures in contaminated data, with implications for compressed sensing, collaborative filtering, and related matrix-analytic tasks.

Abstract

We consider robust low rank matrix estimation as a trace regression when outputs are contaminated by adversaries. The adversaries are allowed to add arbitrary values to arbitrary outputs. Such values can depend on any samples. We deal with matrix compressed sensing, including lasso as a partial problem, and matrix completion, and then we obtain sharp estimation error bounds. To obtain the error bounds for different models such as matrix compressed sensing and matrix completion, we propose a simple unified approach based on a combination of the Huber loss function and the nuclear norm penalization, which is a different approach from the conventional ones. Some error bounds obtained in the present paper are sharper than the past ones.

Paper Structure

This paper contains 46 sections, 28 theorems, 227 equations.

Table of Contents

  1. Introduction
  2. Results
  3. Adversarial matrix compressed sensing
  4. Adversarial estimation of sparse linear regression coefficient (adversarial lasso)
  5. Adversarial matrix completion
  6. Related works
  7. Recent development of robust estimation
  8. Adversarial matrix compressed sensing and adversarial lasso
  9. Sparse coefficient estimation under existence of outliers
  10. Sparse estimation under heavy-tailed distribution
  11. SubGaussian estimator
  12. Dependency between $\{\mathbf{x}_i\}_{i=1}^n$ and $\{\xi_i\}_{i=1}^n$
  13. Adversarial matrix completion
  14. Other topics
  15. Main theorem, key propositions, lemmas and corollaries
  16. ...and 31 more sections

Key Result

Theorem 2.1

Suppose that Assumption a:mcs holds. Consider the optimization problem obj2-intro-m. Suppose that $\lambda_o\sqrt{n} \geq 72 L^4 \sigma$ and $\lambda_* = c_{mcs} \times\lambda_o\sqrt{n}\times L \times r_{\lambda_*}$, where and where $c_{mcs}$ is some numerical constant and $c_\kappa = \frac{c_0+1}{\kappa}$ and $\rho^2$ is the maximum diagnonal element of $\Sigma$. Let where $c_{mcs}'$ is some nu

Theorems & Definitions (67)

  • Definition 2.1: MRE Condition
  • Definition 2.2: $L$-subGaussian random vector
  • Remark 2.1
  • Definition 2.3: $\psi_2$-norm
  • Theorem 2.1
  • Remark 2.2
  • Remark 2.3
  • Definition 2.4: RE Condition
  • Theorem 2.2
  • Remark 2.4
  • ...and 57 more