Nyström Error Beyond $M$-Matrices: A Minimal Diagonally Dominant Obstruction

We study the nuclear-norm error of a column-selected Nyström approximation to $K=(L+γI)^{-1}$, where $L$ is symmetric diagonally dominant and $γ>0$. 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.

Source: arXiv:2607.19282v1 - http://arxiv.org/abs/2607.19282v1 PDF: https://arxiv.org/pdf/2607.19282v1 Original Link: http://arxiv.org/abs/2607.19282v1