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We consider a problem of significant practical importance, namely, the reconstruction of a low-rank data matrix from a small subset of its entries. This problem appears in many areas such as collaborative filtering, computer vision and wireless sensor networks. In this paper, we focus on the matrix completion problem in the case when the observed samples are corrupted by noise. We compare the performance...
Let M be an n?? ?? n matrix of rank r ?? n, and assume that a uniformly random subset E of its entries is observed. We describe an efficient algorithm that reconstructs M from |E| = O(r n) observed entries with relative root mean square error RMSE ?? C(??) (nr/|E|)1/2. Further, if r = O(1) and M is sufficiently unstructured, then it can be reconstructed exactly from |E| = O(n log n) entries. This...
How many random entries of an n times nalpha, rank r matrix are necessary to reconstruct the matrix within an accuracy delta? We address this question in the case of a random matrix with bounded rank, whereby the observed entries are chosen uniformly at random. We prove that, for any delta Gt 0, C(r, delta)n observations are sufficient. Finally we discuss the question of reconstructing the matrix...
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