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Recent years have witnessed the popularity of using rank minimization as aregularizer for various signal processing and machine learning problems. Asrank minimization problems are often converted to nuclear norm minimization(NNM) problems, they have to be solved iteratively and each iteration requirescomputing a singular value decomposition (SVD). Therefore, their solutionsuffers from the high computation cost of multiple SVDs. To relieve this issue,we propose using the block Lanczos method to compute the partial SVDs, wherethe principal singular subspaces obtained in the previous iteration are used tostart the block Lanczos procedure. To avoid the expensive reorthogonalizationin the Lanczos procedure, the block Lanczos procedure is performed for only afew steps. Our block Lanczos with warm start (BLWS) technique can be adopted bydifferent algorithms that solve NNM problems. We present numerical results onapplying BLWS to Robust PCA and Matrix Completion problems. Experimentalresults show that our BLWS technique usually accelerates its host algorithm byat least two to three times.

A Block Lanczos with Warm Start Technique for Accelerating Nuclear Norm Minimization Algorithms