A patch framework consists of a bipartite graph between $n$ points and $m$ local views (patches) and the $d$-dimensional local coordinates of the points due to the views containing them. Given a patch framework, we consider the problem of finding a rigid alignment of the views, identified with an element of the product of $m$ orthogonal groups, $\mathbb{O}(d)^m$, that minimizes the alignment error. In the case when the views are noiseless, a perfect alignment exists, resulting in a realization of the points that respects the geometry of the views. The affine rigidity of such realizations, its connection with the overlapping structure of the views, and its consequences in spectral and semidefinite algorithms has been studied in related work [Zha and Zhang; Chaudhary et al.]. In this work, we characterize the non-degeneracy of a rigid alignment, consequently obtaining a characterization of the local rigidity of a realization, and convergence guarantees on Riemannian gradient descent for aligning the views. Precisely, we characterize the non-degeneracy of an alignment of (possibly noisy) local views based on the kernel and positivity of a certain matrix. Thereafter, we work in the noiseless setting. Under a mild condition on the local views, we show that the non-degeneracy and uniqueness of a perfect alignment, up to the action of $\mathbb{O}(d)$, are equivalent to the local and global rigidity of the resulting realization, respectively. This also yields a characterization of the local rigidity of a realization. We also provide necessary and sufficient conditions on the overlapping structure of the noiseless local views for their realizations to be locally/globally rigid. Finally, we focus on the Riemannian gradient descent for aligning the local views and obtain a sufficient condition on an alignment for the algorithm to converge (locally) linearly to it.

翻译：补丁框架由一张二分图（连接$n$个点与$m个局部视图（补丁）$）以及各视图所包含点的$d$维局部坐标构成。给定一个补丁框架，我们研究如何寻找与$m$个正交群乘积$\mathbb{O}(d)^m$中的元素相对应的视图刚性对齐，以最小化对齐误差。在视图无噪声的情况下，存在完美对齐，从而得到尊重视图几何的点实现。相关研究[Zha and Zhang; Chaudhary et al.]已探讨了此类实现的仿射刚性、其与视图重叠结构的联系，以及在谱算法和半定算法中的推论。本文中，我们刻画了刚性对齐的非退化性，进而获得实现局部刚性的判定条件，并给出了黎曼梯度下降法对齐视图的收敛保证。具体而言，我们基于特定矩阵的核与正定性，刻画了（可能含噪声的）局部视图对齐的非退化性。随后，我们聚焦无噪声场景。在局部视图的温和条件下，我们证明了：完美对齐（在$\mathbb{O}(d)$作用下）的非退化性与唯一性分别等价于所得实现的局部刚性与全局刚性。这同时也给出了实现局部刚性的判定条件。此外，我们提出了无噪声局部视图重叠结构需满足的充要条件，以确保其实现具有局部/全局刚性。最后，我们针对对齐局部视图的黎曼梯度下降法展开分析，得到了算法在该对齐处（局部）线性收敛的充分条件。