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Noisy Low-Rank Matrix Completion via Transformed $L_1$ Regularization and its Theoretical Properties

Kun Zhao, Jiayi Wang, Yifei Lou

TL;DR

The paper tackles noisy low-rank matrix completion under general sampling by applying transformed L1 (TL1) regularization to the singular values, bridging between rank minimization and nuclear-norm relaxation via the hyperparameter $a$. It provides nonasymptotic Frobenius-norm error bounds that match nuclear-norm rates when $a$ is large, and establishes rank control results when $a$ is small, under MAR sampling. Theoretical contributions include minimax-optimal (up to log factors) rates and novel nonconvex analysis for TL1, complemented by an ADMM-based algorithm and extensive simulations and real-data experiments demonstrating TL1's robustness and practical guidance for hyperparameter tuning. These results broaden the understanding of nonconvex regularizations in matrix completion and offer a practical, effective alternative to convex relaxations across diverse sampling scenarios.

Abstract

This paper focuses on recovering an underlying matrix from its noisy partial entries, a problem commonly known as matrix completion. We delve into the investigation of a non-convex regularization, referred to as transformed $L_1$ (TL1), which interpolates between the rank and the nuclear norm of matrices through a hyper-parameter $a \in (0, \infty)$. While some literature adopts such regularization for matrix completion, it primarily addresses scenarios with uniformly missing entries and focuses on algorithmic advances. To fill in the gap in the current literature, we provide a comprehensive statistical analysis for the estimator from a TL1-regularized recovery model under general sampling distribution. In particular, we show that when $a$ is sufficiently large, the matrix recovered by the TL1-based model enjoys a convergence rate measured by the Frobenius norm, comparable to that of the model based on the nuclear norm, despite the challenges posed by the non-convexity of the TL1 regularization. When $a$ is small enough, we show that the rank of the estimated matrix remains a constant order when the true matrix is exactly low-rank. A trade-off between controlling the error and the rank is established through different choices of tuning parameters. The appealing practical performance of TL1 regularization is demonstrated through a simulation study that encompasses various sampling mechanisms, as well as two real-world applications. Additionally, the role of the hyper-parameter $a$ on the TL1-based model is explored via experiments to offer guidance in practical scenarios.

Noisy Low-Rank Matrix Completion via Transformed $L_1$ Regularization and its Theoretical Properties

TL;DR

The paper tackles noisy low-rank matrix completion under general sampling by applying transformed L1 (TL1) regularization to the singular values, bridging between rank minimization and nuclear-norm relaxation via the hyperparameter . It provides nonasymptotic Frobenius-norm error bounds that match nuclear-norm rates when is large, and establishes rank control results when is small, under MAR sampling. Theoretical contributions include minimax-optimal (up to log factors) rates and novel nonconvex analysis for TL1, complemented by an ADMM-based algorithm and extensive simulations and real-data experiments demonstrating TL1's robustness and practical guidance for hyperparameter tuning. These results broaden the understanding of nonconvex regularizations in matrix completion and offer a practical, effective alternative to convex relaxations across diverse sampling scenarios.

Abstract

This paper focuses on recovering an underlying matrix from its noisy partial entries, a problem commonly known as matrix completion. We delve into the investigation of a non-convex regularization, referred to as transformed (TL1), which interpolates between the rank and the nuclear norm of matrices through a hyper-parameter . While some literature adopts such regularization for matrix completion, it primarily addresses scenarios with uniformly missing entries and focuses on algorithmic advances. To fill in the gap in the current literature, we provide a comprehensive statistical analysis for the estimator from a TL1-regularized recovery model under general sampling distribution. In particular, we show that when is sufficiently large, the matrix recovered by the TL1-based model enjoys a convergence rate measured by the Frobenius norm, comparable to that of the model based on the nuclear norm, despite the challenges posed by the non-convexity of the TL1 regularization. When is small enough, we show that the rank of the estimated matrix remains a constant order when the true matrix is exactly low-rank. A trade-off between controlling the error and the rank is established through different choices of tuning parameters. The appealing practical performance of TL1 regularization is demonstrated through a simulation study that encompasses various sampling mechanisms, as well as two real-world applications. Additionally, the role of the hyper-parameter on the TL1-based model is explored via experiments to offer guidance in practical scenarios.

Paper Structure

This paper contains 20 sections, 10 theorems, 91 equations, 1 figure, 9 tables, 1 algorithm.

Table of Contents

  1. INTRODUCTION
  2. PRELIMINARIES
  3. Problem Setup
  4. Notations
  5. Model Formulation
  6. THEORETICAL PROPERTIES
  7. Regime 1: when a is large
  8. Regime 2: when a is small
  9. SIMULATION STUDY
  10. Discussion 1.
  11. Discussion 2.
  12. REAL DATA APPLICATIONS
  13. CONCLUSION
  14. Numerical algorithm
  15. Proof of Theorem \ref{['Them1']} and Theorem \ref{['Them3']}
  16. ...and 5 more sections

Key Result

Theorem 1

Suppose Assumptions assump1-assump3 hold, $A_0 \in \mathbb{R}^{m_1 \times m_2}$ is approximately low-rank in the sense that $\|A_0\|_*/\sqrt{m_1m_2} \leq \gamma$ for a constant $\gamma > 0$, and $\|A_0\|_\infty \leq \zeta$ for a constant $\zeta$. Take $\lambda \asymp \frac{(\zeta \vee \sigma)}{\sqrt with probability at least $1-(\kappa+1)/d$, where $\kappa$ is a constant depending on $L$.

Figures (1)

  • Figure 1: Impact of the parameter $a$ in TL1 on the matrix recovery: relative errors (left) and estimated rank (right) with respect to $a$.

Theorems & Definitions (20)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4: Low-rankness
  • Corollary 1
  • Lemma 1
  • proof : Proof of Lemma \ref{['lemma1']}
  • Lemma 2
  • proof : Proof of Lemma \ref{['lemma2']}
  • Lemma 3
  • ...and 10 more