Consistent Shape Maps via Semidefinite Programming

Recent advances in shape matching have shown that jointly optimizing the maps among the shapes in a collection can lead to significant improvements when compared to estimating maps between pairs of shapes in isolation. These methods typically invoke a cycle‐consistency criterion — the fact that compositions of maps along a cycle of shapes should approximate the identity map. This condition regularizes the network and allows for the correction of errors and imperfections in individual maps. In particular, it encourages the estimation of maps between dissimilar shapes by compositions of maps along a path of more similar shapes. In this paper, we introduce a novel approach for obtaining consistent shape maps in a collection that formulates the cycle‐consistency constraint as the solution to a semidefinite program (SDP). The proposed approach is based on the observation that, if the ground truth maps between the shapes are cycle‐consistent, then the matrix that stores all pair‐wise maps in blocks is low‐rank and positive semidefinite. Motivated by recent advances in techniques for low‐rank matrix recovery via semidefinite programming, we formulate the problem of estimating cycle‐consistent maps as finding the closest positive semidefinite matrix to an input matrix that stores all the initial maps. By analyzing the Karush‐Kuhn‐Tucker (KKT) optimality condition of this program, we derive theoretical guarantees for the proposed algorithm, ensuring the correctness of the recovery when the errors in the inputs maps do not exceed certain thresholds. Besides this theoretical guarantee, experimental results on benchmark datasets show that the proposed approach outperforms state‐of‐the‐art multiple shape matching methods.