We introduce a reformulation of regularized low-rank recovery models to take advantage of GPU, multiple CPU, and hybridized architectures. Low-rank recovery often involves nuclear-norm minimization through iterative thresholding of singular values. These models are slow to fit and difficult to parallelize because of their dependence on computing a singular value decomposition at each iteration. Regularized low-rank recovery models also incorporate non-smooth terms to separate structured components (e.g. sparse outliers) from the low-rank component, making these problems more difficult.
Using Burer-Monteiro splitting and marginalization, we develop a smooth, non-convex formulation of regularized low-rank recovery models that can be fit with first-order solvers. We develop a computable certificate of convergence for this non-convex program, and use it to establish bounds on the suboptimality of any point. Using robust principal component analysis (RPCA) as an example, we include numerical experiments showing that this approach is an order-of-magnitude faster than existing RPCA solvers on the GPU. We also show that this acceleration allows new applications for RPCA, including real-time background subtraction and MR image analysis.
Accelerating nuclear-norm regularized low-rank matrix optimization through Burer-Monteiro decomposition
Institute of Statistical Mathematics（统计数学研究所）；Department of Mathematics University of Maryland at College Park（大学数学系）；Institute of Operations Research and Analytics National University of Singapore（运营研究与分析国家大学研究所）；Department of Mathematics and Institute of Operations Research and Analytics National University of Singapore（大学数学系和运营研究与分析国家大学研究所）
Efficient Group Lasso Regularized Rank Regression with Simulation-Based Tuning
Engineering Systems and Design, Singapore University of Technology and Design, Singapore（新加坡科技设计大学工程系统与设计系）；School of Mathematics and Statistics, Henan University（河南大学数学与统计学院）；School of Mathematics and Statistics & Center for Applied Mathematics of Henan Province, Henan University（河南大学数学与统计学院及河南省应用数学中心）
