Subspace projection matrix completion on Grassmann manifold

In this paper, we work on the problem of subspace estimation from random downsamplings of its projection matrix. An optimization problem on the Grassmann manifold is formulated for projection matrix completion, and an iterative gradient descend line-search algorithm on the Grassmann manifold (GGDLS) is proposed to solve such optimization problem. The convergence of the proposed algorithm has theoretical guarantee, and numerical experiments verify that the required sampling number for successful recovery of a rank s projection matrix in ℝ^N×N with probability 1 is 2s(N - s) in the noiseless cases. Compared with some reference algorithms, in the noiseless scenario, the proposed algorithm is very time efficient, and the required sampling number is rather small for successful recovery. In the noisy scenario, the proposed GGDLS is remarkably robust against the noise both under high measurement SNR and low measurement SNR.