On the upper bounds for the matrix spectral norm

Ryapolov Denis, Maxim Rakhuba, Sergey Samsonov, Alexey Naumov

Keywords: spectral norm, randomized numerical linear algebra, Jacobian

Abstract: Estimating the spectral norm (largest singular value) of an implicitly defined matrix using only matrix-vector products is a fundamental problem in numerical linear algebra and machine learning. The challenge is to estimate it from above with few matrix-vector products (matvecs) as precise as possible. In this paper, we propose a new \emph{Counterbalance} estimator, which provides an upper bounds on the norm, and derive bounds on its underestimation probability. We show that it allows for constructing tighter upper bounds for the operator norm of the underlying matrix and fixed underestimation probability, compared to baseline methods.

Submission Number: 11