Near-Optimal Joint Object Matching via Convex Relaxation

Joint matching over a collection of objects aims at aggregating information from a large collection of similar instances (e.g. images, graphs, shapes) to improve maps between pairs of them. Given multiple objects and matches computed between a few object pairs in isolation, the goal is to recover an entire collection of maps that are (1) globally consistent, and (2) close to the provided maps — and under certain conditions provably the ground-truth maps. Despite recent advances on this problem, the best-known recovery guarantees are limited to a small constant barrier — none of the existing methods find theoretical support when more than 50 % of input correspondences are corrupted. Moreover, prior approaches focus mostly on fully similar objects, while it is practically more demanding to match instances that are only partially similar to each other (e.g., different views of a single physical object). In this paper, we propose an algorithm to jointly match multiple objects that exhibit only partial similarities, given a few (possibly highly incomplete) pairwise matches that are densely corrupted. By encoding a consistent partial map collection into a 0-1 semidefinite matrix, we propose to recover the ground-truth maps via a parameter-free convex program called MatchLift, following a spectral method that pre-estimates the total number of distinct elements to be matched. Numerically, this program can be efficiently solved via alternating direction methods of multipliers (ADMM) along with a greedy rounding strategy. Theoretically, MatchLift exhibits near-optimal error-correction ability, i.e. in the asymptotic regime it is guaranteed to work even when a dominant fraction 1−Θ log2 n