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In this paper, a general optimization formulation is proposed for the subspace segmentation by low rank representation via the sparse-prompting quasi-rank function. We prove that, with the clean data from independent linear subspaces, the optimal solution to our optimization formulation not only is the lowest rank but also forms a block-diagonal matrix, which implies that it is reasonable to use any sparse-prompting quasi-rank function as the measure of the low rank in subspace clustering. With the data contaminated by Gaussian noise and/or gross errors, the alternating direction method of multipliers is applied to solving it and every sub-optimization problem has a closed-form optimal solution when the band restricted thresholding operator induced by its corresponding sparse-prompting function has an analytic expression, in which the gross errors part is replaced with the sparse-prompting matrix function. Finally, taking a specific sparse-prompting function, the fraction function, we conduct a series of simulations on different databases to get the performance of our algorithm tested, and experimental results show that our algorithm can obtain lower clustering error rate and higher value of evaluation indicators ACC, NMI and ARI than other state-of-the-art subspace clustering algorithms on different databases.

Subspace Segmentation by Low Rank Representation via the Sparse-Prompting Quasi-Rank Function