Input-Sparsity Low Rank Approximation in Schatten Norm

Yi Li, David Woodruff
Proceedings of the 37th International Conference on Machine Learning, PMLR 119:6001-6009, 2020.

Abstract

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 $\|A-YZ^T\|_p \leq (1+\eps)\|A-A_k\|_p$, where $\|M\|_p = \left (\sum_{i=1}^n \sigma_i(M)^p \right )^{1/p}$ 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 $\tilde{O}(\nnz(A) + n^{\alpha_p}\poly(k/\eps))$, 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 $\tilde{O}(\nnz(A) + n \cdot \poly(k/\epsilon))$ 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.

Cite this Paper

BibTeX
@InProceedings{pmlr-v119-li20q, title = {Input-Sparsity Low Rank Approximation in Schatten Norm}, author = {Li, Yi and Woodruff, David}, booktitle = {Proceedings of the 37th International Conference on Machine Learning}, pages = {6001--6009}, year = {2020}, editor = {III, Hal Daumé and Singh, Aarti}, volume = {119}, series = {Proceedings of Machine Learning Research}, month = {13--18 Jul}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v119/li20q/li20q.pdf}, url = {https://proceedings.mlr.press/v119/li20q.html}, abstract = {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 $\|A-YZ^T\|_p \leq (1+\eps)\|A-A_k\|_p$, where $\|M\|_p = \left (\sum_{i=1}^n \sigma_i(M)^p \right )^{1/p}$ 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 $\tilde{O}(\nnz(A) + n^{\alpha_p}\poly(k/\eps))$, 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 $\tilde{O}(\nnz(A) + n \cdot \poly(k/\epsilon))$ 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.} }
Endnote
%0 Conference Paper %T Input-Sparsity Low Rank Approximation in Schatten Norm %A Yi Li %A David Woodruff %B Proceedings of the 37th International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2020 %E Hal Daumé III %E Aarti Singh %F pmlr-v119-li20q %I PMLR %P 6001--6009 %U https://proceedings.mlr.press/v119/li20q.html %V 119 %X 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 $\|A-YZ^T\|_p \leq (1+\eps)\|A-A_k\|_p$, where $\|M\|_p = \left (\sum_{i=1}^n \sigma_i(M)^p \right )^{1/p}$ 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 $\tilde{O}(\nnz(A) + n^{\alpha_p}\poly(k/\eps))$, 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 $\tilde{O}(\nnz(A) + n \cdot \poly(k/\epsilon))$ 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.
APA
Li, Y. & Woodruff, D.. (2020). Input-Sparsity Low Rank Approximation in Schatten Norm. Proceedings of the 37th International Conference on Machine Learning, in Proceedings of Machine Learning Research 119:6001-6009 Available from https://proceedings.mlr.press/v119/li20q.html.

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