Randomized low-rank approximation for symmetric indefinite matrices

TL;DR

This paper develops a robust variant of the Nyström method for low-rank approximation of symmetric indefinite matrices, overcoming instability issues present in the standard approach and providing theoretical error bounds.

Contribution

It introduces a new stable algorithm for approximating symmetric indefinite matrices and proves relative-error bounds under certain spectral decay conditions.

Findings

The proposed method is more stable than traditional Nyström for indefinite matrices.

Theoretical error bounds are established for matrices with rapidly decaying singular values.

Experimental results demonstrate the robustness of the new algorithm.

Abstract

The Nystr\"om method is a popular choice for finding a low-rank approximation to a symmetric positive semi-definite matrix. The method can fail when applied to symmetric indefinite matrices, for which the error can be unboundedly large. In this work, we first identify the main challenges in finding a Nystr\"om approximation to symmetric indefinite matrices. We then prove the existence of a variant that overcomes the instability, and establish relative-error nuclear norm bounds of the resulting approximation that hold when the singular values decay rapidly. The analysis naturally leads to a practical algorithm, whose robustness is illustrated with experiments.