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The paper presents several results that address a fundamental question in low-rank matrices recovery: how many measurements are needed to recover low rank matrices? We begin by investigating the complex matrices case and show that $4nr-4r^2$ generic measurements are both necessary and sufficient for the recovery of rank-$r$ matrices in $\C^{n\times n}$ by algebraic tools.
J. F. Cai, E.J. Candès, Z.W. Shen, A singular value thresholding algorithm for matrix completion, SIAM J. Optim., 20(4)(2008), 1956-1982
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Joe Harris, Algebraic geometry
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Radu Balan, Pete Casazza and Dan Edidin, On signal reconstruction without phase, Applied and Computational Harmonic Analysis Volume 20, Issue 3, May 2006, Pages 345¨C356
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E. J. Candès and B. Recht, Exact matrix completion via convex optimization, Foundations of Computational Mathematics 9(6), 717-772 (2009)
2009
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E. J. Candès, Y. Plan, Tight Oracle Inequalities for Low-Rank Matrix Recovery From a Minimal Number of Noisy Random Measurements, IEEE Trans. Inform. Theory 57(4)(2010)2342-2359
2010
Cited alongside, same era.
B. Recht, M. Fazel, and P. Parrilo, Guaranteed minimum-rank solutions of linear matrix equations via nuclear norm minimization, SIAM Rev. 52(3)(2010)471-501
2010
Cited alongside, same era.
P. Parrilo, A. Willsky, V. Chandrasekaran, and B. Recht, The Convex Geometry of Linear Inverse Problems, Foundations of Computational Mathematics December 2012, Volume 12, Issue 6, pp 805-849
2012
Cited alongside, same era.
Y. C. Eldar, D. Needell, Y. Plan, Uniqueness conditions for low-rank matrix recovery, Appl. Comp. Harm. Anal, 33(2013)309-314
2013
Cited alongside, same era.
Cited in the paper.
Cited in the paper.
T. Tony Cai, Anru Zhang, ROP: Matrix recovery via rank-one projections, arXiv:1310.5791
Cited in the paper.
Cited in the paper.
Cited in the paper.
Friedrich Philipp, Phase Retrieval from 4N-4 Measurements, arXiv:1403.4769
Cited in the paper.
Cynthia Vinzant, A small frame and a certificate of its injectivity, arXiv:1502.04656
Cited in the paper.
Afonso S. Bandeira, Jameson Cahill, Dustin G. Mixon, Aaron A. Nelson, Saving phase: Injectivity and stability for phase retrieval, Applied and Computational Harmonic Analysis Volume 37, Issue 1, July 2014, Pages 106-125
2014
Later among the works it cites.
B. G. Bodmann and N. Hammen, Stable phase retrieval with low-redundancy frame
2015
Closest in time.
Aldo Conca, Dan Edidin, Milena Hering and Cynthia Vinzant, An algebraic characterization of injectivity in phase retrieval, Applied and Computational Harmonic Analysis, Volume 38, Issue 2, March 2015, Pages 346-356
2015
Closest in time.
V.Y.F. Tan, L. Balzano and S.C. Draper, Rank Minimization Over Finite Fields: Fundamental Limits and Coding-Theoretic Interpretations, IEEE Transactions on Information Theory, 58(4)(2012)2018-2039
2039
Closest in time.
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