AI 中文总结
研究对\(K=(L+\gamma I)^{-1}\)列选择的Nyström逼近核范数误差是否收益递减，通过舒尔补恒等式简化问题，构造单参数SDD族确定失败区间，证明相关维度情况，推导公式并举例，完整回答了西蒙斯研讨会报告中的问题4.6。
AI 中文摘要
我们研究了对\(K=(L+\gamma I)^{-1}\)进行列选择的Nyström逼近的核范数误差，其中\(L\)是对称对角占优且\(\gamma>0\)。核心问题是该误差是否有收益递减的情况。一个舒尔补恒等式将问题简化为主子矩阵逆的迹。现有的M矩阵结果解决了\(L\)是对称对角占优M矩阵（SDDM）的情况。但仅对角占优是不够的，三维时就会出现失败情况。我们构造了一个精确的单参数SDD族并确定其精确的失败区间。通过一个\(2\times2\)恒等式证明三维是SDD类中的最小维度。接着表明在严格对角占优下失败情况持续存在；对于非空选定基集，四维是最小维度。最后，我们证明了符号切换下的不变性，推导了一个三维公式展示带符号三角形如何导致失败，并给出一个贪婪列选择错过最优对的例子。这些发现完整回答了最近西蒙斯研讨会报告中的问题4.6。
英文摘要
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.
