SOLVING SYSTEMS OF QUADRATIC EQUATIONS VIA EXPONENTIAL-TYPE GRADIENT DESCENT ALGORITHM

• 来源期刊: 计算数学（英文版）
• 关键词: Low-rank matrix recovery Non-convex optimization Phase retrieval
• 年，卷（期）: 2020,（4）
• 页码范围: 638-660
• 页数: 23页
• 语种: 英文
• DOI: 10.4208/jcm.1902-m2018-0109

摘要

We consider the rank minimization problem from quadratic measurements,i.e.,recovering a rank r matrix X ∈ Rn×r from m scalar measurements yi =aTiXXT ai,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) =1/4m Σmi=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 (nr log(cr)) Gaussian random measurements.