SIAM Journal on Optimization, Volume 31, Issue 1, Page 1049-1077, January 2021.
We consider the local sensitivity of least-squares formulations of inverse problems. The sets of inputs and outputs of these problems are assumed to have the structures of Riemannian manifolds. The problems we consider include the approximation problem of finding the nearest point on a Riemannian embedded submanifold from a given point in the ambient space. We characterize the first-order sensitivity, i.e., condition number, of local minimizers and critical points to arbitrary perturbations of the input of the least-squares problem. This condition number involves the Weingarten map of the input manifold, which measures the amount by which the input manifold curves in its ambient space. We validate our main results through experiments with the $n$-camera triangulation problem in computer vision.

中文翻译：

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黎曼近似问题的条件数

SIAM优化杂志，第31卷，第1期，第1049-1077页，2021年1月。
我们考虑反问题的最小二乘公式的局部敏感性。假定这些问题的输入和输出集具有黎曼流形的结构。我们考虑的问题包括一个近似问题，即要从环境空间中的给定点找到黎曼嵌入子流形上的最近点。我们刻画了局部最小化子和临界点对最小二乘问题输入的任意扰动的一阶灵敏度，即条件数。此条件编号涉及输入歧管的Weingarten映射，该映射测量输入歧管在其环境空间中弯曲的数量。我们通过对计算机视觉中的$ n $-相机三角剖分问题进行实验来验证我们的主要结果。