Local Rigidity of Patch Framework Realization

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.