Fast Length-Squared Sampling for Positive-Semidefinite Matrices

Motivated by an application in machine learning optimization, this paper focuses on the challenges of sampling a matrix uniformly from the unit spectral norm ball. It is proven that all singular values of sampled matrices converge to 1 almost surely as the matrix dimensions increase. This result provides the theoretical justification for a proposed simple sampling method applicable for large dimension sizes matching matrices found in modern large language models. Experimental results demonstrate both the convergence of the singular values, as well as the exact and proposed approximate sampling methods.

The power method is one of the most fundamental tools for extracting top principal components from data through low-rank matrix approximation. Yet, when the target rank is large, the cost of matrix multiplication associated with this procedure becomes a major bottleneck. We develop an algorithmic and theoretical framework for accelerating the power method using fast sketching, which is a popular paradigm in randomized linear algebra. Our framework leads to simple and provably efficient methods for singular value decomposition, low-rank factorization, and Nyström approximation, which attain strong numerical performance on benchmark problems. The key novelty in our analysis is the use of regularized spectral approximation, a property of fast sketching methods which proves more flexible in generalizing power method guarantees than traditional arguments.

The $2 \rightarrow q$ norm of a matrix $X \in \mathbb{R}^{n \times d}$ is defined as $\lVert X \rVert_{2 \rightarrow q} = \sup_{\lVert v \rVert_2 = 1} \lVert Xv \rVert_q$. We give polynomial-time multiplicative approximation algorithms for this norm when $q > 2$ (i.e. in the hypercontractive setting). This problem either directly captures or is closely related to long-standing open problems in combinatorial optimization and hardness of approximation (e.g. Small Set Expansion), quantum information (e.g. Best Separable State), and algorithmic statistics. Very little is known about what approximation factors we can achieve for this problem in polynomial time, even though such approximations have significant downstream consequences. Barak, Brandão, Harrow, Kelner, Steurer, and Zhou showed that no polynomial-time algorithm can achieve an approximation factor better than $2^{\sqrt{\log n}}$, assuming the Exponential Time Hypothesis (FOCS'12). On the other hand, a simple spectral algorithm gives a $d^{1/4}$-approximation as a baseline. We give, to the best of our knowledge, the first polynomial-time approximation algorithm beating this baseline by polynomial factors. For the important special case of $q = 4$ it achieves a $d^{1/8}$-approximation. All previous algorithms required additional assumptions on $X$, or only surpassed the baseline for small values of $n$. Moreover, we construct sum-of-squares certificates for the $2 \rightarrow q$ norm. This directly implies improved algorithms for robust mean and covariance estimation, robust regression, and clustering, when the data only satisfies a bound on its $q$-th moment.