Noisy matrix decomposition via convex relaxation: Optimal rates in high dimensions (1102.4807v3)

Abstract: We analyze a class of estimators based on convex relaxation for solving high-dimensional matrix decomposition problems. The observations are noisy realizations of a linear transformation $\mathfrak{X}$ of the sum of an approximately) low rank matrix $\Theta^\star$ with a second matrix $\Gamma^\star$ endowed with a complementary form of low-dimensional structure; this set-up includes many statistical models of interest, including factor analysis, multi-task regression, and robust covariance estimation. We derive a general theorem that bounds the Frobenius norm error for an estimate of the pair $(\Theta^\star, \Gamma^\star)$ obtained by solving a convex optimization problem that combines the nuclear norm with a general decomposable regularizer. Our results utilize a "spikiness" condition that is related to but milder than singular vector incoherence. We specialize our general result to two cases that have been studied in past work: low rank plus an entrywise sparse matrix, and low rank plus a columnwise sparse matrix. For both models, our theory yields non-asymptotic Frobenius error bounds for both deterministic and stochastic noise matrices, and applies to matrices $\Theta^\star$ that can be exactly or approximately low rank, and matrices $\Gamma^\star$ that can be exactly or approximately sparse. Moreover, for the case of stochastic noise matrices and the identity observation operator, we establish matching lower bounds on the minimax error. The sharpness of our predictions is confirmed by numerical simulations.

• The paper presents a convex optimization framework for noisy matrix decomposition that achieves optimal non-asymptotic Frobenius error bounds.
• It leverages a spikiness condition to relax singular vector incoherence requirements, broadening its applicability to low-rank and sparse models.
• Numerical simulations validate the theoretical predictions, underscoring its potential in robust covariance estimation and multi-task regression.

Noisy Matrix Decomposition via Convex Relaxation: Optimal Rates in High Dimensions

The paper under discussion presents a sophisticated analysis of a class of convex relaxation estimators designed for tackling high-dimensional noisy matrix decomposition problems. This research is relevant for a broad spectrum of statistical models, such as factor analysis, multi-task regression, and robust covariance estimation. The core problem involves decomposing a matrix $Y$ into the sum of a low-rank matrix $\Theta^\star$ and another matrix $\Gamma^\star$ that possesses a complementary low-dimensional structure.

Theoretical Contributions

A significant contribution of this paper is the derivation of an upper bound on the Frobenius norm error for the estimated pair $(\Theta^\star, \Gamma^\star)$ obtained through a convex optimization model coupling the nuclear norm with a decomposable regularizer. The authors impose a "spikiness" condition, which is a milder variant of singular vector incoherence, to achieve these error bounds. Notably, they specialize their outcomes to include scenarios previously explored: low rank with entrywise sparsity and low rank with columnwise sparsity. These scenarios are vital as they extend the applicability of matrix decomposition techniques to both exactly and approximately low-rank and sparse matrices, under deterministic and stochastic noise conditions.

The achievability results hinge upon addressing the curse of dimensionality by obtaining sharp non-asymptotic Frobenius error bounds. Additionally, a remarkable aspect of this paper is the establishment of matching minimax lower bounds, demonstrating the incapability of further improving the results beyond constant factors under the specified noise constraints.

Numerical Simulations and Extended Analysis

The theoretical findings are supported by numerical simulations that validate the theoretical predictions of the error bounds' sharpness, reinforcing the estimators' efficacy under varying conditions of rank and sparsity. Furthermore, the research anticipates practical implementations in robust covariance estimation and multi-task regression by adapting the model to assorted observation operators beyond the identity mapping.

In a comparative discussion with previous works, notably those of Hsu et al., the paper's approaches are distinguished by utilizing a spikiness condition instead of full singular vector incoherence assumptions, leading to broader applicability in the context of noisy observations.

Implications and Future Directions

Practically, the implications of this paper are noteworthy for fields requiring dimensionality reduction while handling large data volumes contaminated with noise. This research theoretically underpins advancements in recommendation systems, image processing, and bioinformatics, where data matrices often exhibit underlying low-rank structures amidst sparse noise.

The paper sets a foundation for future research paths, such as exploring decompositions where both components are constrained by decomposable regularizers, allowing potential expansion into new application domains. Furthermore, adaptations to partial observation models, similar to those in matrix completion, present promising avenues for extending the results to scenarios with constrained data availability.

In conclusion, the research offers a robust theoretical framework for matrix decomposition in high-dimensional spaces, providing both insights into estimator behavior under noise and paving the way for further developments in composite regularizer-based matrix analysis.