Krylov Methods are (nearly) Optimal for Low-Rank Approximation
This work addresses fundamental computational bottlenecks in low-rank approximation for machine learning and numerical linear algebra, resolving open questions and providing near-optimal algorithms.
The paper tackles the problem of rank-1 low-rank approximation under various Schatten norms, showing that Krylov methods nearly achieve the information-theoretically optimal number of matrix-vector products, with bounds such as Ω(log(n)/ε^{1/2}) for Spectral LRA and O(log(1/ε)/ε^{1/3}) for fixed constant p.
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 $Ω\left(\log(n)/\varepsilon^{1/2}\right)$ 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 $Ω\left(1/\varepsilon^{1/3}\right)$ lower bound, to a $\log(1/\varepsilon)$ factor.