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A RankCorrected Procedure for Matrix Completion with Fixed Basis Coefficients
, 2012
"... In this paper, we address lowrank matrix completion problems with fixed basis coefficients, which include the lowrank correlation matrix completion in various fields such as the financial market and the lowrank density matrix completion from the quantum state tomography. For this class of problem ..."
Abstract

Cited by 5 (2 self)
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In this paper, we address lowrank matrix completion problems with fixed basis coefficients, which include the lowrank correlation matrix completion in various fields such as the financial market and the lowrank density matrix completion from the quantum state tomography. For this class of problems, the efficiency of the common nuclear norm penalized estimator for recovery may be challenged. Here, with a reasonable initial estimator, we propose a rankcorrected procedure to generate an estimator of high accuracy and low rank. For this new estimator, we establish a nonasymptotic recovery error bound and analyze the impact of adding the rankcorrection term on improving the recoverability. We also provide necessary and sufficient conditions for rank consistency in the sense of Bach [3], in which the concept of constraint nondegeneracy in matrix optimization plays an important role. As a byproduct, our results provide a theoretical foundation for the majorized penalty method of Gao and Sun [25] and Gao [24] for structured lowrank matrix optimization problems.
MATRIX COMPLETION MODELS WITH FIXED BASIS COEFFICIENTS AND RANK REGULARIZED PROBLEMS WITH HARD CONSTRAINTS
, 2013
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A Semismooth NewtonCG Based Dual PPA for Matrix Spectral Norm Approximation Problems
 MATH. PROGRAM., SER. A
, 2014
"... We consider a class of matrix spectral norm approximation problems for finding an affine combination of given matrices having the minimal spectral norm subject to some prescribed linear equality and inequality constraints. These problems arise often in numerical algebra, engineering and other area ..."
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We consider a class of matrix spectral norm approximation problems for finding an affine combination of given matrices having the minimal spectral norm subject to some prescribed linear equality and inequality constraints. These problems arise often in numerical algebra, engineering and other areas, such as finding Chebyshev polynomials of matrices and fastest mixing Markov chain models. Based on classical analysis of proximal point algorithms (PPAs) and recent developments on