Low-Rank Matrix Recovery From Errors and Erasures

Chen, Yudong; Jalali, Ali; Sanghavi, Sujay; Caramanis, Constantine

doi:10.1109/tit.2013.2249572

Cited by 129 publications

(94 citation statements)

References 27 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…It has been shown [30]- [32] that exact recovery is possible via nuclear norm minimization, as soon as the number of observed entries exceeds the order of the information theoretic limit. This line of algorithms is also robust against noise and outliers [33], [34], and allows exact recovery even in the presence of a constant portion of adversarially corrupted entries [35]- [37], which have found numerous applications in collaborative filtering [38], medical imaging [39], [40], etc. Nevertheless, the theoretical guarantees of these algorithms do not apply to the more structured observation models associated with the proposed multi-fold Hankel structure.…”

Section: B Connection and Comparison To Prior Workmentioning

confidence: 99%

Robust Spectral Compressed Sensing via Structured Matrix Completion

Chen

Chi

2014

IEEE Trans. Inform. Theory

284

443

View full text Add to dashboard Cite

Abstract-The paper explores the problem of spectral compressed sensing, which aims to recover a spectrally sparse signal from a small random subset of its n time domain samples. The signal of interest is assumed to be a superposition of r multidimensional complex sinusoids, while the underlying frequencies can assume any continuous values in the normalized frequency domain. Conventional compressed sensing paradigms suffer from the basis mismatch issue when imposing a discrete dictionary on the Fourier representation. To address this issue, we develop a novel algorithm, called Enhanced Matrix Completion (EMaC), based on structured matrix completion that does not require prior knowledge of the model order. The algorithm starts by arranging the data into a low-rank enhanced form exhibiting multi-fold Hankel structure, and then attempts recovery via nuclear norm minimization. Under mild incoherence conditions, EMaC allows perfect recovery as soon as the number of samples exceeds the order of r log 4 n, and is stable against bounded noise. Even if a constant portion of samples are corrupted with arbitrary magnitude, EMaC still allows exact recovery, provided that the sample complexity exceeds the order of r 2 log 3 n. Along the way, our results demonstrate the power of convex relaxation in completing a low-rank multi-fold Hankel or Toeplitz matrix from minimal observed entries. The performance of our algorithm and its applicability to super resolution are further validated by numerical experiments.

show abstract

Section: B Connection and Comparison To Prior Workmentioning

confidence: 99%

Robust Spectral Compressed Sensing via Structured Matrix Completion

Chen

Chi

2014

IEEE Trans. Inform. Theory

284

443

View full text Add to dashboard Cite

show abstract

“…The work in [31], [32] consider matrix completion -recovering a low-rank matrix from an overwhelming number of erasures. The work initiated in [33], and subsequently continued and extended in [34], [35] focuses on recovering a low-rank matrix from erasures and possibly gross but sparse corruptions. In the noiseless case, stacking all our samples as columns of a p × n matrix, we indeed obtain a corrupted low rank matrix.…”

Section: Organization and Notationmentioning

confidence: 99%

Outlier-Robust PCA: The High-Dimensional Case

Caramanis

Mannor

2013

IEEE Trans. Inform. Theory

Self Cite

107

View full text Add to dashboard Cite

Principal Component Analysis plays a central role in statistics, engineering and science. Because of the prevalence of corrupted data in real-world applications, much research has focused on developing robust algorithms. Perhaps surprisingly, these algorithms are unequipped -indeed, unable -to deal with outliers in the high dimensional setting where the number of observations is of the same magnitude as the number of variables of each observation, and the data set contains some (arbitrarily) corrupted observations. We propose a High-dimensional Robust Principal Component Analysis (HR-PCA) algorithm that is as efficient as PCA, robust to contaminated points, and easily kernelizable. In particular, our algorithm achieves maximal robustness -it has a breakdown point of 50% (the best possible) while all existing algorithms have a breakdown point of zero. Moreover, our algorithm recovers the optimal solution exactly in the case where the number of corrupted points grows sub linearly in the dimension.

show abstract

“…In other words, the L1 model (3.7) is akin to a robust PCA (a.k.a principal component pursuit) for incomplete matrix A, see [3] for analysis and computation of (3.8) and [4] for a related convex model.…”

Section: L1 Modelsmentioning

confidence: 99%

A Coordinate Descent Method for Robust Matrix Factorization and Applications

Sheen¹

2016

SIURO

View full text Add to dashboard Cite

Matrix factorization methods are widely used for extracting latent factors for low rank matrix completion and rating prediction problems arising in recommender systems of on-line retailers. Most of the existing models are based on L2 fidelity (quadratic functions of factorization error). In this work, a coordinate descent (CD) method is developed for matrix factorization under L1 fidelity so that the related minimization is done one variable at a time and the factorization error is sparsely distributed. In low rank random matrix completion and rating prediction of MovieLens-100k datasets, the CDL1 method shows remarkable stability and accuracy under gross corruption of training (observation) data while the L2 fidelity based methods rapidly deteriorate. A closed form analytical solution is found for the one-dimensional L1-fidelity subproblem, and is used as a building block of CDL1 algorithm whose convergence is analyzed. The connection with the well-known convex method, the robust principal component analysis (RPCA), is made. A comparison with RPCA on recovering low rank Gaussian matrices under sparse and independent Gaussian noise shows that CDL1 maintains accuracy at much lower sampling ratios (from much fewer observed entries) than that for RPCA.

show abstract

Low-Rank Matrix Recovery From Errors and Erasures

Cited by 129 publications

References 27 publications

Robust Spectral Compressed Sensing via Structured Matrix Completion

Robust Spectral Compressed Sensing via Structured Matrix Completion

Outlier-Robust PCA: The High-Dimensional Case

A Coordinate Descent Method for Robust Matrix Factorization and Applications

Contact Info

Product

Resources

About