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arXiv · 2607.19282

Nystr\"om Error Beyond $M$-Matrices: A Minimal Diagonally Dominant Obstruction

Matthew J. Colbrook
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Abstract

We study the nuclear-norm error of a column-selected Nystr\"om approximation to $K=(L+\gamma I)^{-1}$, where $L$ is symmetric diagonally dominant and $\gamma>0$. Our central question is whether this error has diminishing returns. A Schur-complement identity reduces the question to traces of inverses of principal submatrices. Existing $M$-matrix results settle the case in which $L$ is a symmetric diagonally dominant $M$-matrix (SDDM). However, diagonal dominance alone is not enough: failure occurs already in dimension three. We construct an exact one-parameter SDD family and determine its sharp failure interval. A $2\times2$ identity proves that dimension three is minimal within the SDD class. We then show that failure persists under strict diagonal dominance; with a nonempty selected base set, dimension four is minimal. Finally, we prove invariance under signature switching, derive a three-dimensional formula showing how a signed triangle causes failure, and give an example in which greedy column selection misses the optimal pair. Together, these findings complete the answer to Problem 4.6 in a recent Simons workshop report.

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Matthew J. Colbrook. 2026-07-21. Nystr\"om Error Beyond $M$-Matrices: A Minimal Diagonally Dominant Obstruction. https://arxiv.org/abs/2607.19282

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