Randomized biorthogonalization through a two-sided Gram-Schmidt process

Abstract

We propose and analyze a randomized two-sided Gram-Schmidt process for the biorthogonalization of two given matrices $X, Y \in\mathbb{R}^{n\times m}$. The algorithm aims to find two matrices $Q, P \in\mathbb{R}^{n\times m}$ such that ${\rm range}(X) = {\rm range}(Q)$, ${\rm range}(Y) = {\rm range}(P)$ and $(\Omega Q)^T \Omega P = I$, where $\Omega \in\mathbb{R}^{s \times n}$ is a sketching matrix satisfying an oblivious subspace $\varepsilon$-embedding property; in other words, the biorthogonality condition on the columns of $Q$ and $P$ is replaced by an equivalent condition on their sketches. This randomized approach is computationally less expensive than the classical two-sided Gram-Schmidt process, has better numerical stability, and the condition number of the computed bases $Q, P$ is often smaller than in the deterministic case. Several different implementations of the randomized algorithm are analyzed and compared numerically. The randomized two-sided Gram-Schmidt process is applied to the nonsymmetric Lancozs algorithm for the approximation of eigenvalues and both left and right eigenvectors.