Adaptive decoupled strategy for robust and efficient low-rank matrix decomposition

Abstract

Low-rank matrix decomposition with missing values is vital in computer vision and pattern recognition, yet it presents significant challenges. This problem can be viewed as a separable nonlinear optimization, but traditional methods often fail to account for the coupling between parameters and the impact of solution properties on visual reconstruction. We observe that such separable nonlinear problems frequently encounters narrow ravines filled with sharp minima. Classic alternating optimization methods, the Wiberg algorithm and its variants tend to linger in these regions, converging to sharp minima, thereby slowing convergence and degrading reconstruction quality. This promotes us to introduce the Adaptive Decoupled Variable Projection algorithm (ADVP), which can adaptively handle the coupling of parameters, significantly accelerate the convergence rate, and dynamically adjust the parameter search subspace, helping algorithms avoid these ravines towards flatter local minima. These flat minima exhibit robustness against missing data, noise, and outliers, enhancing the quality of visual reconstruction. Extensive experiments on synthetic and real datasets have validated the efficiency of ADVP and its superior performance in visual reconstruction.