Krylov Methods are (nearly) Optimal for Low-Rank Approximation

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 $\ellp$ 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.