On solving a rank regularized minimization problem via equivalent factorized column-sparse regularized models

報告題目：On solving a rank regularized minimization problem via equivalent factorized column-sparse regularized models

報 告 人：邊偉 教授  哈爾濱工業(yè)大學(xué) 杰青

邀請人：趙志華  

報告時間：2025年4月22日（周二） 16:00-18:15

報告地點：北校區(qū)會議中心104

報告人簡介：邊偉，哈爾濱工業(yè)大學(xué)數(shù)學(xué)學(xué)院，教授、博士生導(dǎo)師。2004年和2009年于哈爾濱工業(yè)大學(xué)分別獲得學(xué)士和博士學(xué)位。2010-2012年訪問香港理工大學(xué)，跟隨陳小君教授從事博士后工作。主要研究領(lǐng)域為：最優(yōu)化理論與算法。先后在 Math. Program., Math. Oper. Res., SIAM J. Optim., SIAM J. Numer. Anal., SIAM J. Sci. Comput., SIAM J. Imaging Sci. 等期刊發(fā)表多篇學(xué)術(shù)論文。先后獲國家級青年人才稱號和國家級人才稱號。現(xiàn)任SCI期刊Journal of Optimization Theory and Application編委，中國運籌學(xué)會常務(wù)理事，黑龍江省數(shù)學(xué)會常務(wù)理事，中國運籌學(xué)會數(shù)學(xué)規(guī)劃分會理事等。

報告摘要：Rank regularized minimization problem is an ideal model for the low-rank matrix completion/recovery problem. The matrix factorization approach can transform the high-dimensional rank regularized problem to a low-dimensional factorized column sparse regularized problem. The latter can greatly facilitate fast computations in applicable algorithms, but needs to overcome the simultaneous non-convexity of the loss and regularization functions. In this paper, we consider the factorized column sparse regularized model. Firstly, we optimize this model with bound constraints, and establish a certain equivalence between the optimized factorization problem and rank regularized problem. Further, we strengthen the optimality condition for stationary points of the factorization problem and define the notion of strong stationary point. Moreover, we establish the equivalence between the factorization problem and its non-convex relaxation in the sense of global minimizers and strong stationary points. To solve the factorization problem, we design two types of algorithms and give an adaptive method to reduce their computation. The first algorithm is from the relaxation point of view and its iterates own some properties from global minimizers of the factorization problem after finite iterations. We give some analysis on the convergence of its iterates to a strong stationary point. The second algorithm is designed for directly solving the factorization problem. We improve the PALM algorithm introduced by Bolte et al. (Math Program Ser A 146:459–494, 2014) for the factorization problem and give its improved convergence results. Finally, we conduct numerical experiments to show the promising performance of the proposed model and algorithms for low-rank matrix completion.

Joint work with Wenjing Li and Kim-Chuan Toh

主辦單位：數(shù)學(xué)與統(tǒng)計學(xué)院