Low-Rank Representation Code Implementation with IADM-NNLS Algorithm

Application Background: This code implements IADM_NNLS (Inexact Alternating Direction Method for Nuclear Norm Regularized Least Squares), specifically designed for solving optimization problems with nuclear norm regularization terms like low-rank representation algorithms. The method improves matrix rank determination by not only ensuring it remains less than or equal to the number of rows but also providing more accurate calculation of the maximum number of linearly independent vectors in a set. The algorithm alternates between solving nuclear norm minimization and least squares subproblems using proximal operators. Key Technical Implementation: The core technique involves determining the size of the maximum linearly independent vector set through matrix rank computation. From a definition perspective, for a vector set A, the rank corresponds to the cardinality of its maximum linearly independent subset. The implementation uses singular value thresholding (SVT) for nuclear norm minimization and employs accelerated gradient descent for the least squares component. Key functions include: - Nuclear norm proximal operator using singular value decomposition - Inexact ADM convergence criteria with adaptive penalty parameters - Linear independence testing through rank-revealing QR factorization This approach enables more accurate identification of vector dependencies and enhances solution stability for rank-constrained optimization problems.