[Paper Review] Strongly Convex Programming for Exact Matrix Completion and Robust Principal Component Analysis

Hui Zhang, Jian‐Feng Cai+2 more|arXiv (Cornell University)|Dec 16, 2011

Sparse and Compressive Sensing Techniques31 references3 citations

TL;DR

This paper establishes that strongly convex programming formulations can exactly recover low-rank matrices in both matrix completion and robust principal component analysis under suitable conditions. It provides explicit lower bounds for the regularization parameter τ that guarantee exact recovery, extending classical nuclear norm minimization results to more stable, well-conditioned optimization models.

ABSTRACT

The common task in matrix completion (MC) and robust principle component analysis (RPCA) is to recover a low-rank matrix from a given data matrix. These problems gained great attention from various areas in applied sciences recently, especially after the publication of the pioneering works of Cand`es et al.. One fundamental result in MC and RPCA is that nuclear norm based convex optimizations lead to the exact low-rank matrix recovery under suitable conditions. In this paper, we extend this result by showing that strongly convex optimizations can guarantee the exact low-rank matrix recovery as well. The result in this paper not only provides sufficient conditions under which the strongly convex models lead to the exact low-rank matrix recovery, but also guides us on how to choose suitable parameters in practical algorithms.

Motivation & Objective

• To extend the theoretical guarantees of nuclear norm minimization in low-rank matrix recovery to strongly convex optimization models.
• To derive sufficient conditions under which strongly convex formulations achieve exact low-rank matrix recovery in matrix completion (MC) and robust PCA (RPCA).
• To provide explicit, data-dependent lower bounds for the regularization parameter τ in strongly convex models to ensure exact recovery.
• To guide practical algorithm design by identifying suitable parameter choices that maintain exact recovery guarantees.

Proposed method

• Formulates matrix completion and RPCA as strongly convex optimization problems by adding a quadratic regularization term to the nuclear norm or 1-norm objectives.
• Derives sufficient conditions on the regularization parameter τ to ensure that the true low-rank matrix is the unique solution to the strongly convex program.
• Uses probabilistic arguments and matrix concentration inequalities to bound the operator and infinity norms of the data and residual matrices.
• Applies the Cauchy-Schwarz inequality and matrix projection properties to estimate the lower bound of τ in terms of observable data norms.
• Establishes that the lower bound for τ depends on the infinity norm and Frobenius norm of the observed data matrix D.
• Adapts existing recovery results from convex optimization (e.g., [9]) to the strongly convex case by verifying that key assumptions still hold under the new formulation.

Experimental results

Research questions

• RQ1 Can strongly convex relaxations of the nuclear norm and 1-norm minimization problems still guarantee exact low-rank matrix recovery in matrix completion and RPCA?
• RQ2 What are the sufficient conditions on the regularization parameter τ that ensure exact recovery in strongly convex formulations?
• RQ3 How can the lower bound for τ be estimated using only the observed data matrix D in practice?
• RQ4 Does the strongly convex formulation preserve the exact recovery guarantees of classical convex optimization models?

Key findings

• Strongly convex programming formulations can exactly recover low-rank matrices in both matrix completion and robust PCA under the same conditions as classical convex optimization.
• A sufficient lower bound for the regularization parameter τ is derived as τ ≥ (2‖D‖∞ + (λ√15/3)‖D‖F) / (λ(1−λ)), which ensures exact recovery with high probability.
• The derived lower bound for τ is computable from the observed data matrix D alone, making it practical for algorithm design.
• The results are complementary to prior work on convex nuclear norm minimization and extend its theoretical guarantees to well-conditioned, strongly convex models.
• The analysis confirms that the key assumptions from [9] remain valid under the strongly convex formulation, allowing direct adaptation of existing recovery proofs.