Schatten范数下的输入稀疏低秩近似
Input-Sparsity Low Rank Approximation in Schatten Norm
摘要：
我们给出了秩$k$低秩近似问题在每个Schatten范数下的第一输入稀疏时间算法。具体而言，对于给定的$n\times n$矩阵$A$，我们的算法计算$Y,Z\in \mathbb{R}^{n\times k}$，它以高概率满足$\|A-YZ^T\|_p \leq (1+ε)\|A-A_k\|_p$，其中$\|M\|_p = \left (\sum_{i=1}^n σ_i(M)^p \right )^{1/p}$是具有奇异值$σ_1(M), \ldots, σ_n(M)$的矩阵$M$的Schatten $p$ -范数，并且其中$A_k$是$A$的最佳秩- $k$近似.我们的算法在时间上运行$\tilde{O}(\operatorname{nnz}(A) + mn^{α_p}\operatorname{poly}(k/ε))$，其中$α_p = 0$用于$p\in [1,2)$，$α_p = (ω-1)(1-2/p)$用于$p>2$，$ω\approx 2.374$是矩阵乘法的指数。对于重要的情况下$p = 1$，这对应于更“强大”的核规范，我们得到$\tilde{O}(\operatorname{nnz}(A) + m \cdot \operatorname{poly}(k/ε))$时间，这是以前只知道Frobenius规范（$p = 2$）。此外，由于每个$p$的$α_p < ω- 1$，我们的算法有一个更好的依赖于$n$比奇异值分解的每个$p$。对我们的分析至关重要的是使用Ky-Fan $p$ -norms的降维。
英文摘要：
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 $Y,Z\in \mathbb{R}^{n\times k}$, which, with high probability, satisfy $\|A-YZ^T\|_p \leq (1+ε)\|A-A_k\|_p$, where $\|M\|_p = \left (\sum_{i=1}^n σ_i(M)^p \right )^{1/p}$ is the Schatten $p$-norm of a matrix $M$ with singular values $σ_1(M), \ldots, σ_n(M)$, and where $A_k$ is the best rank-$k$ approximation to $A$. Our algorithm runs in time $\tilde{O}(\operatorname{nnz}(A) + mn^{α_p}\operatorname{poly}(k/ε))$, where $α_p = 0$ for $p\in [1,2)$ and $α_p = (ω-1)(1-2/p)$ for $p>2$ and $ω\approx 2.374$ is the exponent of matrix multiplication. For the important case of $p = 1$, which corresponds to the more "robust" nuclear norm, we obtain $\tilde{O}(\operatorname{nnz}(A) + m \cdot \operatorname{poly}(k/ε))$ time, which was previously only known for the Frobenius norm ($p = 2$). Moreover, since $α_p < ω- 1$ 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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
