We give the first input-sparsity time algorithms for the rank-k low rank approximation problem in every Schatten norm. Specifically, for a given n\times n matrix A, our algorithm computes Y,Z\in R^{n\times k}, which, with high probability, satisfy \|A-YZ^T\|_p \leq (1+\epsilon)\|A-A_k\|_p, where \|M\|_p = \left (i=1^n \sigma_i(M)^p \right )^{1/p} is the Schatten p-norm of a matrix M with singular values \sigma_1(M), \ldots, \sigma_n(M), and where A_k is the best rank-k approximation to A. Our algorithm runs in time O(nnz(A) + n ^{\alpha_p}poly(k/\epsilon)), where \alpha_p = 1 for p\in [1,2) and \alpha_p = 1 + (\omega-1)(1-2/p) for p>2 and \omega \approx 2.374 is the exponent of matrix multiplication. For the important case of p = 1, which corresponds to the more "robust" nuclear norm, we obtain O(nnz(A) + n \cdot poly(k/\epsilon)) time, which was previously only known for the Frobenius norm (p = 2). Moreover, since \alpha_p < \omega  for every p, our algorithm has a better dependence on n than that in the singular value decomposition for every p. Crucial to our analysis is the use of dimensionality reduction for Ky-Fan p-norms.
We give the first input-sparsity time algorithms for the rank-k low rank approximation problem in every Schatten norm. Specifically, for a given n\times n matrix A, our algorithm computes Y,Z\in R^{n\times k}, which, with high probability, satisfy \|A-YZ^T\|_p \leq (1+\epsilon)\|A-A_k\|_p, where \|M\|_p = \left (i=1^n \sigma_i(M)^p \right )^{1/p} is the Schatten p-norm of a matrix M with singular values \sigma_1(M), \ldots, \sigma_n(M), and where A_k is the best rank-k approximation to A. Our algorithm runs in time O(nnz(A) + mn ^{\alpha_p}poly(k/\epsilon)), where \alpha_p = 0 for p\in [1,2) and \alpha_p = (\omega-1)(1-2/p) for p>2 and \omega \approx 2.374 is the exponent of matrix multiplication. For the important case of p = 1, which corresponds to the more "robust" nuclear norm, we obtain O(nnz(A) + m \cdot poly(k/\epsilon)) time, which was previously only known for the Frobenius norm (p = 2). Moreover, since \alpha_p < \omega - 1 for every p, our algorithm has a better dependence on n than that in the singular value decomposition for every p. Crucial to our analysis is the use of dimensionality reduction for Ky-Fan p-norms.
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