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Low-Rank Positive Semidefinite Matrix Recovery From Corrupted Rank-One Measurements

Yuanxin Li, Yue Sun, Yuejie Chi
2017 IEEE Transactions on Signal Processing  

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2016 pre-print arXiv:1602.02737v1 fatcat:wh57chvcfzgk7gjmkfqttnxj5i

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2016 accepted
arXiv:1602.02737v2 fatcat:ggqo7wxmbrffnj2fknszu67piy
We study the problem of estimating a low-rank positive semidefinite (PSD) matrix from a set of rank-one measurements using sensing vectors composed of i.i.d. standard Gaussian entries, which are possibly  ...  We establish that with high probability, a low-rank PSD matrix can be exactly recovered as soon as the number of measurements is large enough, even when a fraction of the measurements are corrupted by  ...  INTRODUCTION In many emerging applications of science and engineering, we are interested in estimating a low-rank positive semidefinite (PSD) matrix X 0 ∈ R n×n from a set of magnitude measurements that  ... 
doi:10.1109/tsp.2016.2620109 fatcat:62e4lgyq3zeapd7axnxcz7b7su
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Yuanxin Li, Yue Sun, Yuejie Chi. "Low-Rank Positive Semidefinite Matrix Recovery From Corrupted Rank-One Measurements." IEEE Transactions on Signal Processing 65.2 (2017) 397-408

Outlier-robust recovery of low-rank positive semidefinite matrices from magnitude measurements

Yue Sun, Yuanxin Li, Yuejie Chi
2016 2016 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)  

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We address the problem of estimating a low-rank positive semidefinite (PSD) matrix from a set of magnitude measurements that are quadratic in the sensing vectors in the presence of arbitrary outliers.  ...  It is shown that the algorithm can exactly recover a rank-r PSD matrix of size-n from O nr 2 measurements with high probability, even when a fraction of the measurements is corrupted by arbitrary outliers  ...  In this paper, we focus on robust recovery of the low-rank PSD matrix when the measurements are further corrupted by outliers, which are possibly adversarial with arbitrary amplitudes.  ... 
doi:10.1109/icassp.2016.7472442 dblp:conf/icassp/SunLC16 fatcat:ug3wwy4yozd2dftguhighzx53a
Citation

Yue Sun, Yuanxin Li, Yuejie Chi. "Outlier-robust recovery of low-rank positive semidefinite matrices from magnitude measurements." 2016 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP) (2016) 4069-4073

PhaseLift is robust to a constant fraction of arbitrary errors

Paul Hand
2017 Applied and Computational Harmonic Analysis  

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This nonconvex problem may be convexified into a semidefinite rank-one matrix recovery problem, known as PhaseLift.  ...  From this perspective, the proposed convex program is simpler that the semidefinite version of the sparse-plus-low-rank formulation standard in the robust PCA literature.  ...  Relation to Robust PCA Much recent work in matrix completion has studied the recovery of low-rank matrices from arbitrary corruptions to its entries.  ... 
doi:10.1016/j.acha.2016.01.001 fatcat:otpgzjbyq5hsrb4qki5jhtpr2y
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Paul Hand. "PhaseLift is robust to a constant fraction of arbitrary errors." Applied and Computational Harmonic Analysis 42.3 (2017) 550-562

PhaseLift is robust to a constant fraction of arbitrary errors [article]

Paul Hand
2015 arXiv   pre-print

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Through the technique of lifting, this nonconvex problem may be convexified into a semidefinite rank-one matrix recovery problem, known as PhaseLift.  ...  From this perspective, the proposed convex program is simpler that the semidefinite version of the sparse-plus-low-rank formulation standard in the robust PCA literature.  ...  As in [7, 2] , the positive semidefinite cone provides enough of a constraint to enforce low-rankness.  ... 
arXiv:1502.04241v1 fatcat:v6hqiolsffekhhq52defp7rfji
Citation

Paul Hand. "PhaseLift is robust to a constant fraction of arbitrary errors." arXiv (2015)

Denise: Deep Robust Principal Component Analysis for Positive Semidefinite Matrices [article]

Calypso Herrera, Florian Krach, Anastasis Kratsios, Pierre Ruyssen, Josef Teichmann
2021 arXiv   pre-print

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2020 pre-print arXiv:2004.13612v2 fatcat:xqzo3jqo7jan3d6lvkjxffnety

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2020 pre-print
arXiv:2004.13612v1 fatcat:k44iz6jxfzgs7kbn5bxwx6fdiu
The currently available methods for performing such a low-rank plus sparse decomposition are matrix specific, meaning, those algorithms must re-run for every new matrix.  ...  Therefore, we introduce Denise, a deep learning-based algorithm for robust PCA of covariance matrices, or more generally of symmetric positive semidefinite matrices, which learns precisely such a function  ...  the low-rank+sparse decomposition of an example matrix from both datasets.  ... 
arXiv:2004.13612v3 fatcat:vpelmlgwdjehrgfy2ry7dd6mba
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Calypso Herrera, Florian Krach, Anastasis Kratsios, Pierre Ruyssen, Josef Teichmann. "Denise: Deep Robust Principal Component Analysis for Positive Semidefinite Matrices." arXiv (2021)

Stable rank one matrix completion is solved by two rounds of semidefinite programming relaxation [article]

Augustin Cosse, Laurent Demanet
2017 arXiv   pre-print

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This paper studies the problem of deterministic rank-one matrix completion.  ...  In this paper, we show that in every instance where the problem has a unique solution, one can provably recover the original matrix through two rounds of semidefinite programming relaxation with minimization  ...  Both authors were supported by a grant from the MISTI MIT-Belgium seed fund. AC was supported by the FNRS, FSMP, BAEF and Francqui Foundations.  ... 
arXiv:1801.00368v1 fatcat:uu4hbt6kwjcf3nh3lmbckcygua
Citation

Augustin Cosse, Laurent Demanet. "Stable rank one matrix completion is solved by two rounds of semidefinite programming relaxation." arXiv (2017)

Nonconvex Robust Low-rank Matrix Recovery [article]

Xiao Li, Zhihui Zhu, Anthony Man-Cho So, Rene Vidal
2019 arXiv   pre-print

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2018 pre-print arXiv:1809.09237v3 fatcat:nx4qklc6u5fupatvsmwoeomdri

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2018 pre-print
arXiv:1809.09237v1 fatcat:kznyixwbabgsxalza5fi7r7jua
2018 pre-print
arXiv:1809.09237v2 fatcat:momcoujf2zdhxa4dsnum34do7i
In this paper we study the problem of recovering a low-rank matrix from a number of random linear measurements that are corrupted by outliers taking arbitrary values.  ...  We demonstrate the efficacy of the SubGM for the nonconvex robust low-rank matrix recovery problem with various numerical experiments.  ...  Nonconvex Robust Low-Rank Matrix Recovery: Symmetric Positive Semidefinite Case.  ... 
arXiv:1809.09237v4 fatcat:52gwc3ymsnbz5l6cfhlpfsskvy
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Xiao Li, Zhihui Zhu, Anthony Man-Cho So, Rene Vidal. "Nonconvex Robust Low-rank Matrix Recovery." arXiv (2019)

Deterministic Symmetric Positive Semidefinite Matrix Completion

William E. Bishop, Byron M. Yu
2014 Neural Information Processing Systems  

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We consider the problem of recovering a symmetric, positive semidefinite (SPSD) matrix from a subset of its entries, possibly corrupted by noise.  ...  We develop a set of sufficient conditions for the recovery of a SPSD matrix from a set of its principal submatrices, present necessity results based on this set of conditions and develop an algorithm that  ...  Introduction There are multiple scenarios where we might wish to reconstruct a symmetric positive semidefinite (SPSD) matrix from a sampling of its entries.  ... 
dblp:conf/nips/BishopY14 fatcat:u4zfoaderfac3e633gnbrfhe6q
Citation

William E. Bishop, Byron M. Yu. "Deterministic Symmetric Positive Semidefinite Matrix Completion." Neural Information Processing Systems (2014) 2762-2770

The Stability of Low-Rank Matrix Reconstruction: A Constrained Singular Value View

Gongguo Tang, Arye Nehorai
2012 IEEE Transactions on Information Theory  

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The stability of low-rank matrix reconstruction with respect to noise is investigated in this paper.  ...  Isotropic and subgaussian measurement operators are shown to have -CMSVs bounded away from zero with high probability, as long as the number of measurements is relatively large.  ...  Suppose is a matrix of rank ; the low-rank matrix reconstruction problem aims at recovering matrix from a set of linear measurements corrupted by noise (1) where is a linear measurement operator.  ... 
doi:10.1109/tit.2012.2204535 fatcat:3hpum5mfsfebnh2jr7lvo224tq
Citation

Gongguo Tang, Arye Nehorai. "The Stability of Low-Rank Matrix Reconstruction: A Constrained Singular Value View." IEEE Transactions on Information Theory 58.9 (2012) 6079-6092

Rank-One Measurements of Low-Rank PSD Matrices Have Small Feasible Sets [article]

T. Mitchell Roddenberry, Santiago Segarra, Anastasios Kyrillidis
2021 arXiv   pre-print

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arXiv:2012.09768v1 fatcat:saavw3r4jbegbducryi6uytafi
We study the role of the constraint set in determining the solution to low-rank, positive semidefinite (PSD) matrix sensing problems.  ...  The setting we consider involves rank-one sensing matrices: In particular, given a set of rank-one projections of an approximately low-rank PSD matrix, we characterize the radius of the set of PSD matrices  ...  Introduction We study the recovery of low-rank and approximately low-rank matrices from linear measurements.  ... 
arXiv:2012.09768v2 fatcat:wigccx5yt5ek7kd6seiwkxvvx4
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T. Mitchell Roddenberry, Santiago Segarra, Anastasios Kyrillidis. "Rank-One Measurements of Low-Rank PSD Matrices Have Small Feasible Sets." arXiv (2021)

Two proposals for robust PCA using semidefinite programming

Michael McCoy, Joel A. Tropp
2011 Electronic Journal of Statistics  

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2010 pre-print
arXiv:1012.1086v2 fatcat:d3ema4serfaerd3ds4fprlommq
2010 accepted
arXiv:1012.1086v3 fatcat:mq6wrmmv2fcc7ewln534cbqyji
2010 pre-print
arXiv:1012.1086v1 fatcat:wrz3urzitfcgngpz7tp3ljukym
The second method produces a low-leverage decomposition (LLD) of the data that attempts to form a low-rank model for the data by separating out corrupted observations.  ...  This paper proposes two novel approaches for robust PCA based on semidefinite programming.  ...  . , T. by a low-rank model while the remainder come from another population or are corrupted by measurement noise.  ... 
doi:10.1214/11-ejs636 fatcat:eovfpae4lndpxe7an3z2w6ihie
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Michael McCoy, Joel A. Tropp. "Two proposals for robust PCA using semidefinite programming." Electronic Journal of Statistics 5.0 (2011) 1123-1160

A Certifiably Correct Algorithm for Synchronization over the Special Euclidean Group [article]

David M. Rosen, Luca Carlone, Afonso S. Bandeira, John J. Leonard
2017 arXiv   pre-print

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2016 pre-print
arXiv:1611.00128v1 fatcat:o7hfsnzc2jecbmhsi7a7jq5ygq
2016 pre-print
arXiv:1611.00128v2 fatcat:62lpvuo3bjfdvh2xsv5touqh2q
We develop a specialized optimization scheme for solving large-scale instances of this semidefinite relaxation by exploiting its low-rank, geometric, and graph-theoretic structure to reduce it to an equivalent  ...  Experimental evaluation on a variety of simulated and real-world pose-graph SLAM datasets shows that SE-Sync is capable of recovering globally optimal solutions when the available measurements are corrupted  ...  The resulting rank-restricted form of the problem is thus a low-dimensional nonlinear program, rather than a semidefinite program.  ... 
arXiv:1611.00128v3 fatcat:idsjd5mezjapxflvs5t4rn6m74
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David M. Rosen, Luca Carlone, Afonso S. Bandeira, John J. Leonard. "A Certifiably Correct Algorithm for Synchronization over the Special Euclidean Group." arXiv (2017)

3-D motion recovery via low rank matrix restoration on articulation graphs

Kun Li, Meiyuan Wang, Yu-Kun Lai, Jingyu Yang, Feng Wu
2017 2017 IEEE International Conference on Multimedia and Expo (ICME)  

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We propose a novel low-rank solution that effectively integrates both a low-rank model for robust skeleton recovery based on temporal coherence, and an articulation-graphbased isometric constraint for  ...  This paper addresses the challenge of 3-D skeleton recovery by exploiting the spatio-temporal correlations of corrupted 3-D skeleton sequences. A skeleton sequence is represented as a matrix.  ...  [11] achieve smooth 3-D skeleton recovery via low-rank matrix recovery.  ... 
doi:10.1109/icme.2017.8019486 dblp:conf/icmcs/LiWLYW17 fatcat:tqzsevyf3zcapcwtmue32la64e
Citation

Kun Li, Meiyuan Wang, Yu-Kun Lai, Jingyu Yang, Feng Wu. "3-D motion recovery via low rank matrix restoration on articulation graphs." 2017 IEEE International Conference on Multimedia and Expo (ICME) (2017) 721-726

Bilinear low-rank coding framework and extension for robust image recovery and feature representation

Zhao Zhang, Shuicheng Yan, Mingbo Zhao, Fan-Zhang Li
2015 Knowledge-Based Systems  

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We mainly study the low-rank image recovery problem by proposing a bilinear low-rank coding framework called Tensor Low-Rank Representation.  ...  For enhanced low-rank recovery and error correction, our method constructs a low-rank tensor subspace to reconstruct given images along row and column directions simultaneously by computing two low-rank  ...  Based on the low-rank recovery to the original data with corruptions and errors corrected, we construct the entries W LR i;j of W LR by using the cosine similarity-style measure as W LR i;j ¼ W LR j;i  ... 
doi:10.1016/j.knosys.2015.06.001 fatcat:ggik3r5jafcqpjezziu3igd2ke
Citation

Zhao Zhang, Shuicheng Yan, Mingbo Zhao, Fan-Zhang Li. "Bilinear low-rank coding framework and extension for robust image recovery and feature representation." Knowledge-Based Systems (2015) 143-157

Robust Principal Component Analysis: Exact Recovery of Corrupted Low-Rank Matrices via Convex Optimization

John Wright, Arvind Ganesh, Shankar R. Rao, YiGang Peng, Yi Ma
2009 Neural Information Processing Systems  

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This paper considers the idealized "robust principal component analysis" problem of recovering a low rank matrix A from corrupted observations D = A + E.  ...  A by-product of our analysis is the first proportional growth results for the related problem of completing a low-rank matrix from a small fraction of its entries.  ...  matrix A from highly corrupted measurements D = A + E.  ... 
dblp:conf/nips/WrightGRPM09 fatcat:6qkyb57llvhx7dugckgj6nzone
Citation

John Wright, Arvind Ganesh, Shankar R. Rao, YiGang Peng, Yi Ma. "Robust Principal Component Analysis: Exact Recovery of Corrupted Low-Rank Matrices via Convex Optimization." Neural Information Processing Systems (2009) 2080-2088

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