Year: 2020
Author: Meng Huang, Zhiqiang Xu
Journal of Computational Mathematics, Vol. 38 (2020), Iss. 4 : pp. 638–660
Abstract
We consider the rank minimization problem from quadratic measurements, i.e., recovering a rank r matrix X∈Rn×r from m scalar measurements yi=aTiXXTai,ai∈Rn,i=1,…,m. Such problem arises in a variety of applications such as quadratic regression and quantum state tomography. We present a novel algorithm, which is termed exponential−type gradient descent algorithm, to minimize a non-convex objective function f(U)=14m∑mi=1(yi−aTiUUTai)2. This algorithm starts with a careful initialization, and then refines this initial guess by iteratively applying exponential-type gradient descent. Particularly, we can obtain a good initial guess of X as long as the number of Gaussian random measurements is O(nr), and our iteration algorithm can converge linearly to the true X (up to an orthogonal matrix) with m=O(nrlog(cr)) Gaussian random measurements.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/jcm.1902-m2018-0109
Journal of Computational Mathematics, Vol. 38 (2020), Iss. 4 : pp. 638–660
Published online: 2020-01
AMS Subject Headings:
Copyright: COPYRIGHT: © Global Science Press
Pages: 23
Keywords: Low-rank matrix recovery Non-convex optimization Phase retrieval.
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