Non-degenerate Rigid Alignment in a Patch Framework

Abstract: Given a set of overlapping local views (patches) of a dataset, we consider the problem of finding a rigid alignment of the views that minimizes a $2$-norm based alignment error. In general, the views are noisy and a perfect alignment may not exist. In this work, we characterize the non-degeneracy of an alignment in the noisy setting based on the kernel and positivity of a certain matrix. This leads to a polynomial time algorithm for testing the non-degeneracy of a given alignment. Subsequently, we focus on Riemannian gradient descent for minimizing the alignment error, providing a sufficient condition on an alignment for the algorithm to converge (locally) linearly to it. \revadd{Additionally, we provide an exact recovery and noise stability analysis of the algorithm}. In the case of noiseless views, a perfect alignment exists, resulting in a realization of the points that respects the geometry of the views. Under a mild condition on the views, we show that a non-degenerate perfect alignment \revadd{characterizes the infinitesimally rigidity of a realization, and thus the local rigidity of a generic realization}. By specializing the non-degeneracy conditions to the noiseless case, we derive necessary and sufficient conditions on the overlapping structure of the views for \revadd{a perfect alignment to be non-degenerate and equivalently, for the resulting realization to be infinitesimally rigid}. Similar results are also derived regarding the uniqueness of a perfect alignment and global rigidity.