A Gradient Descent Algorithm on the Grassman Manifold for Matrix Completion
Provides an efficient method for matrix completion, a key problem in collaborative filtering and data reconstruction, with strong empirical performance.
The paper presents OptSpace, an algorithm for low-rank matrix completion that uses SVD initialization followed by local optimization on the Grassmann manifold, achieving exact reconstruction from a very small subset of entries and demonstrating robustness to noise.
We consider the problem of reconstructing a low-rank matrix from a small subset of its entries. In this paper, we describe the implementation of an efficient algorithm called OptSpace, based on singular value decomposition followed by local manifold optimization, for solving the low-rank matrix completion problem. It has been shown that if the number of revealed entries is large enough, the output of singular value decomposition gives a good estimate for the original matrix, so that local optimization reconstructs the correct matrix with high probability. We present numerical results which show that this algorithm can reconstruct the low rank matrix exactly from a very small subset of its entries. We further study the robustness of the algorithm with respect to noise, and its performance on actual collaborative filtering datasets.