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.

翻译：Nyström方法是寻找对称半正定矩阵低秩近似的一种常用方法。当应用于对称不定矩阵时，该方法可能失效，其误差可能无界增大。本文首先识别了寻找对称不定矩阵的Nyström近似时面临的主要挑战。随后我们证明了存在一种变体能够克服此不稳定性，并在奇异值快速衰减的情况下建立了所得近似的相对误差核范数界。该分析自然地导出了一种实用算法，并通过实验验证了其鲁棒性。