TY - JOUR
T1 - The stability of low-rank matrix reconstruction
T2 - A constrained singular value view
AU - Tang, Gongguo
AU - Nehorai, Arye
N1 - Funding Information:
Manuscript received June 20, 2010; revised October 01, 2011; accepted March 23, 2012. Date of publication June 13, 2012; date of current version August 14, 2012. This work was supported in part by the Office of Naval Research under Grant N000140810849 and in part by the National Science Foundation under Grants CCF-1014908 and CCF-0963742.
PY - 2012
Y1 - 2012
N2 - The stability of low-rank matrix reconstruction with respect to noise is investigated in this paper. The ℓ*-constrained minimal singular value (ℓ*-CMSV) of the measurement operator is shown to determine the recovery performance of nuclear norm minimization-based algorithms. Compared with the stability results using the matrix restricted isometry constant, the performance bounds established using ℓ*-CMSV are more concise, and their derivations are less complex. Isotropic and subgaussian measurement operators are shown to have ℓ*-CMSVs bounded away from zero with high probability, as long as the number of measurements is relatively large. The ℓ*-CMSV for correlated Gaussian operators are also analyzed and used to illustrate the advantage of ℓ*-CMSV compared with the matrix restricted isometry constant. We also provide a fixed point characterization of ℓ*-CMSV that is potentially useful for its computation.
AB - The stability of low-rank matrix reconstruction with respect to noise is investigated in this paper. The ℓ*-constrained minimal singular value (ℓ*-CMSV) of the measurement operator is shown to determine the recovery performance of nuclear norm minimization-based algorithms. Compared with the stability results using the matrix restricted isometry constant, the performance bounds established using ℓ*-CMSV are more concise, and their derivations are less complex. Isotropic and subgaussian measurement operators are shown to have ℓ*-CMSVs bounded away from zero with high probability, as long as the number of measurements is relatively large. The ℓ*-CMSV for correlated Gaussian operators are also analyzed and used to illustrate the advantage of ℓ*-CMSV compared with the matrix restricted isometry constant. We also provide a fixed point characterization of ℓ*-CMSV that is potentially useful for its computation.
KW - correlated design
KW - matrix Dantzig selector (mDS)
KW - matrix LASSO estimator (mLASSO)
KW - matrix basis pursuit (mBP)
KW - restricted isometry property
KW - ℓ-constrained minimal singular value (CMSV)
UR - https://www.scopus.com/pages/publications/84865400442
U2 - 10.1109/TIT.2012.2204535
DO - 10.1109/TIT.2012.2204535
M3 - Article
AN - SCOPUS:84865400442
SN - 0018-9448
VL - 58
SP - 6079
EP - 6092
JO - IEEE Transactions on Information Theory
JF - IEEE Transactions on Information Theory
IS - 9
M1 - 6217312
ER -