We consider the problem of rank-$1$ low-rank approximation (LRA) in the matrix-vector product model under various Schatten norms: $$ \min_{\|u\|_2=1} \|A (I - u u^\top)\|_{\mathcal{S}_p} , $$ where $\|M\|_{\mathcal{S}_p}$ denotes the $\ell_p$ norm of the singular values of $M$. Given $\varepsilon>0$, our goal is to output a unit vector $v$ such that $$ \|A(I - vv^\top)\|_{\mathcal{S}_p} \leq (1+\varepsilon) \min_{\|u\|_2=1}\|A(I - u u^\top)\|_{\mathcal{S}_p}. $$ Our main result shows that Krylov methods (nearly) achieve the information-theoretically optimal number of matrix-vector products for Spectral ($p=\infty$), Frobenius ($p=2$) and Nuclear ($p=1$) LRA. In particular, for Spectral LRA, we show that any algorithm requires $\Omega\left(\log(n)/\varepsilon^{1/2}\right)$ matrix-vector products, exactly matching the upper bound obtained by Krylov methods [MM15, BCW22]. Our lower bound addresses Open Question 1 in [Woo14], providing evidence for the lack of progress on algorithms for Spectral LRA and resolves Open Question 1.2 in [BCW22]. Next, we show that for any fixed constant $p$, i.e. $1\leq p =O(1)$, there is an upper bound of $O\left(\log(1/\varepsilon)/\varepsilon^{1/3}\right)$ matrix-vector products, implying that the complexity does not grow as a function of input size. This improves the $O\left(\log(n/\varepsilon)/\varepsilon^{1/3}\right)$ bound recently obtained in [BCW22], and matches their $\Omega\left(1/\varepsilon^{1/3}\right)$ lower bound, to a $\log(1/\varepsilon)$ factor.

翻译：我们研究在矩阵-向量乘积模型下，针对各种Schatten范数的秩-$1$低秩近似（LRA）问题：$$ \min_{\|u\|_2=1} \|A (I - u u^\top)\|_{\mathcal{S}_p} , $$ 其中$\|M\|_{\mathcal{S}_p}$表示$M$的奇异值的$\ell_p$范数。给定$\varepsilon>0$，我们的目标是输出一个单位向量$v$，使得 $$ \|A(I - vv^\top)\|_{\mathcal{S}_p} \leq (1+\varepsilon) \min_{\|u\|_2=1}\|A(I - u u^\top)\|_{\mathcal{S}_p}. $$ 我们的主要结果表明，对于谱范数（$p=\infty$）、Frobenius范数（$p=2$）和核范数（$p=1$）的LRA，Krylov方法（几乎）达到了信息论最优的矩阵-向量乘积次数。特别地，对于谱范数LRA，我们证明任何算法都需要$\Omega\left(\log(n)/\varepsilon^{1/2}\right)$次矩阵-向量乘积，这与Krylov方法[MM15, BCW22]获得的上界精确匹配。我们的下界回答了[Woo14]中的开放问题1，为谱范数LRA算法缺乏进展提供了证据，并解决了[BCW22]中的开放问题1.2。接下来，我们证明对于任意固定常数$p$，即$1\leq p=O(1)$，存在一个上界$O\left(\log(1/\varepsilon)/\varepsilon^{1/3}\right)$次矩阵-向量乘积，这意味着复杂度不随输入规模增长。这改进了[BCW22]最近获得的$O\left(\log(n/\varepsilon)/\varepsilon^{1/3}\right)$上界，并与它们的$\Omega\left(1/\varepsilon^{1/3}\right)$下界匹配，相差一个$\log(1/\varepsilon)$因子。