Non-negative matrix factorization (NMF) minimizes the euclidean distance between the data matrix and its low rank approximation, and it fails when applied to corrupted data because the loss function is sensitive to outliers. In this paper, we propose a Truncated CauchyNMF loss that handle outliers by truncating large errors, and develop a Truncated CauchyNMF to robustly learn the subspace on noisy datasets contaminated by outliers. We theoretically analyze the robustness of Truncated CauchyNMF comparing with the competing models and theoretically prove that Truncated CauchyNMF has a generalization bound which converges at a rate of order @math , where @math is the sample size. We evaluate Truncated CauchyNMF by image clustering on both simulated and real datasets. The experimental results on the datasets containing gross corruptions validate the effectiveness and robustness of Truncated CauchyNMF for learning robust subspaces.
Zhang . @cite_17 assumed that the dataset contains both Laplace distributed noise and Gaussian distributed noise and proposed an @math -norm regularized Robust NMF (RNMF- @math ) as follows: @math , where @math is a positive constant that trades off the sparsity of @math . Similar to @math -NMF, RNMF- @math is also less sensitive to outliers than NMF, but they are both non-robust to large numbers of outliers because the @math -minimization model has a low breakdown point. Moreover, it is non-trivial to determine the tradeoff parameter @math .
@cite_17
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The Noisy Non-negative Matrix factorization (NMF) is: given a data matrix A (d × n), find non-negative matrices B, C (d × k, k × n respy.) so that A = BC + N, where N is a noise matrix. Existing polynomial time algorithms with proven error guarantees require each column N.,j to have l1 norm much smaller than ||(BC).,j||1, which could be very restrictive. In important applications of NMF such as Topic Modeling as well as theoretical noise models (eg. Gaussian with high σ), almost every column of N.j violates this condition. We introduce the heavy noise model which only requires the average noise over large subsets of columns to be small. We initiate a study of Noisy NMF under the heavy noise model. We show that our noise model subsumes noise models of theoretical and practical interest (for eg. Gaussian noise of maximum possible σ). We then devise an algorithm TSVDNMF which under certain assumptions on B,C, solves the problem under heavy noise. Our error guarantees match those of previous algorithms. Our running time of O((n + d)2k) is substantially better than the O(n3d) for the previous best. Our assumption on B is weaker than the "Separability" assumption made by all previous results. We provide empirical justification for our assumptions on C. We also provide the first proof of identifiability (uniqueness of B) for noisy NMF which is not based on separability and does not use hard to check geometric conditions. Our algorithm outperforms earlier polynomial time algorithms both in time and error, particularly in the presence of high noise.