论文标题
Schatten Norm中的输入 - 标准级别近似
Input-Sparsity Low Rank Approximation in Schatten Norm
论文作者
论文摘要
我们在每个Schatten Norm中给出了排名-K $低等级近似问题的第一个输入 - 标准时间算法。 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}^nσ_i（m）^p \ right）^{1/p} $是矩阵$ m $的schatten $ p $ - 带有单数值$σ_1（m），\ ldots，\ ldots，$ kn（m）$ nim $ nim nim nim an和a $ a $ a $ a $ a $ a nim a $ a $ a $ a $ a $ $ a $。我们的算法在时间上运行$ \ tilde {o}（\ operatorAtorName {nnz}（a） + mn^{α_p} \ propatorAtorName {poly}（k/ε））$ $ω\大约2.374 $是矩阵乘法的指数。对于$ p = 1 $的重要情况，与更“强大的”核定常相对应，我们获得了$ \ tilde {o}（\ propatatorName {nnz}（a） + m \ cdot \ cdot \ cdot \ cdot \ properatorname {poly}（poly}（k/ε））$，以前仅以Frobenius Norm（$ p = 2 $）而闻名。此外，由于每个$ p $ $α_p<ω-1 $，因此我们的算法对$ n $的依赖性比每$ p $的单数值分解中都具有更好的依赖性。对我们的分析至关重要的是使用降低维数$ p $ norms的使用。
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.
