AI 中文总结
本文研究投影扰动下D-最优传感器布置的运动稳定性，提出基于瓶颈匹配距离的裕度条件，证明小扰动下最优解保持稳定，并给出反例说明仅依赖特征间隙的界限不足。
AI 中文摘要
在时变D-最优子集选择中，被替换坐标的数量并不衡量物理位移：当一组移动传感器的最优选择发生变化时，单个传感器可能不得不穿越整个图。因此，我们通过固定移动图上的瓶颈匹配距离来衡量运动，并将半径为P处的距离裕度定义为与当前最大化器距离超过P的配置中最小的旧目标损失。我们的主要结果表明，如果投影仪漂移较小，且所选半径处的距离裕度超过一个显式阈值，则新目标的每个最大化器都位于该半径内，无需平局打破规则或对子集施加均匀条件假设。此外，我们表明这种分离信息确实是必需的：一个加权路径构造表明，在固定的基线特征间隙下，任意小的扰动可以将最优传感器从图的一端移动到另一端，因此仅依赖于扰动大小和特征间隙的界限不能随扰动消失。我们还给出了一个有限证书，即局部和Voronoi限制保持全局最优的条件，以及过采样和正则化扩展，并指出了结果适用的学习字典标准。
英文摘要
In time-varying D-optimal subset selection, the number of replaced coordinates does not measure physical displacement: when the optimal selection for a set of mobile sensors changes, a single sensor may have to cross the entire graph. We therefore measure movement by a bottleneck matching distance on a fixed mobility graph, and define the distance margin at radius P as the smallest old-objective loss among configurations farther than P from the current maximizer. Our main result shows that if the projector drift is small and the distance margin at a chosen radius exceeds an explicit threshold, then every maximizer of the new objective lies within that radius, without tie-breaking rules or uniform conditioning assumptions over subsets. Moreover, we show that separation information of this kind is genuinely needed: a weighted-path construction shows that, at a fixed baseline eigengap, an arbitrarily small perturbation can move an optimal sensor from one end of the graph to the other, so no bound depending only on perturbation size and eigengap can vanish with the perturbation. We also give a finite certificate, conditions under which local and Voronoi restrictions retain a global optimum, and oversampled and regularized extensions, and we identify the learned dictionary criteria to which the results apply.
Comments21 pages, 2 images
