ArticleSIAM journal on optimization : a publication of the Society for Industrial and Applied Mathematics2020
NOISY MATRIX COMPLETION: UNDERSTANDING STATISTICAL GUARANTEES FOR CONVEX RELAXATION VIA NONCONVEX OPTIMIZATION.
Article in SIAM journal on optimization : a publication of the Society for Industrial and Applied Mathematics, 2020. The graph could read no effect estimate from its abstract, so it casts no vote on the map. Cited by 11 papers.
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11 citing papers in PubMed.
- Matrix Completion When Missing Is Not at Random and Its Applications in Causal Panel Data Models.Journal of the American Statistical Association · 2024Article
- Approximate message passing from random initialization with applications toProceedings of the National Academy of Sciences of the United States of America · 2023Article
- Convex and Nonconvex Optimization Are Both Minimax-Optimal for Noisy Blind Deconvolution under Random Designs.Journal of the American Statistical Association · 2023Article
- GNMR: A Provable One-Line Algorithm for Low Rank Matrix Recovery.SIAM journal on mathematics of data science · 2022Article
- Sample-Efficient Reinforcement Learning for Linearly-Parameterized MDPs with a Generative Model.Advances in neural information processing systems · 2021Article
- BRIDGING CONVEX AND NONCONVEX OPTIMIZATION IN ROBUST PCA: NOISE, OUTLIERS, AND MISSING DATA.Annals of statistics · 2021Article
- A SHRINKAGE PRINCIPLE FOR HEAVY-TAILED DATA: HIGH-DIMENSIONAL ROBUST LOW-RANK MATRIX RECOVERY.Annals of statistics · 2021Article
- A selective overview of deep learning.Statistical science : a review journal of the Institute of Mathematical Statistics · 2021Article
- ASYMMETRY HELPS: EIGENVALUE AND EIGENVECTOR ANALYSES OF ASYMMETRICALLY PERTURBED LOW-RANK MATRICES.Annals of statistics · 2021Article
- Comment: Feature Screening and Variable Selection via Iterative Ridge Regression.Technometrics : a journal of statistics for the physical, chemical, and engineering sciences · 2020Article
- Inference and uncertainty quantification for noisy matrix completion.Proceedings of the National Academy of Sciences of the United States of America · 2019Article
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5 authors.
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Abstract
This paper studies noisy low-rank matrix completion: given partial and noisy entries of a large low-rank matrix, the goal is to estimate the underlying matrix faithfully and efficiently. Arguably one of the most popular paradigms to tackle this problem is convex relaxation, which achieves remarkable efficacy in practice. However, the theoretical support of this approach is still far from optimal in the noisy setting, falling short of explaining its empirical success. We make progress towards demystifying the practical efficacy of convex relaxation vis-à-vis random noise. When the rank and the condition number of the unknown matrix are bounded by a constant, we demonstrate that the convex programming approach achieves near-optimal estimation errors - in terms of the Euclidean loss, the entrywise loss, and the spectral norm loss - for a wide range of noise levels. All of this is enabled by bridging convex relaxation with the nonconvex Burer-Monteiro approach, a seemingly distinct algorithmic paradigm that is provably robust against noise. More specifically, we show that an approximate critical point of the nonconvex formulation serves as an extremely tight approximation of the convex solution, thus allowing us to transfer the desired statistical guarantees of the nonconvex approach to its convex counterpart.
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