DS++: A flexible, scalable and provably tight relaxation for matching problems

Abstract

Correspondence problems are often modelled as quadratic optimization problems over permutations. Common scalable methods for approximating solutions of these NP-hard problems are the spectral relaxation for non-convex energies and the doubly stochastic (DS) relaxation for convex energies. Lately, it has been demonstrated that semidefinite programming relaxations can have considerably improved accuracy at the price of a much higher computational cost.We present a convex quadratic programming relaxation which is provably stronger than both DS and spectral relaxations, with the same scalability as the DS relaxation. The derivation of the relaxation also naturally suggests a projection method for achieving meaningful integer solutions which improves upon the standard closest-permutation projection. Our method can be easily extended to optimization over doubly stochastic matrices, partial or injective matching, and problems with additional linear constraints. We employ recent advances in optimization of linear-assignment type problems to achieve an efficient algorithm for solving the convex relaxation.We present experiments indicating that our method is more accurate than local minimization or competing relaxations for non-convex problems. We successfully apply our algorithm to shape matching and to the problem of ordering images in a grid, obtaining results which compare favorably with state of the art methods.We believe our results indicate that our method should be considered the method of choice for quadratic optimization over permutations.

Partial Recovery in the Graph Alignment Problem

“…A second step consequently involves the projection of the doubly stochastic matrix obtained to the set of permutation matrices, a projection that can be done in polynomial time, via, e.g., the Hungarian algorithm. Quadratic programming relaxations of this type are studied in [22,47,26,16,20,21,51]. These relaxations sometimes also involve a quadratic regularization term added to the objective, which corresponds to the Frobenius norm of the doubly stochastic matrix.…”

A comprehensive survey on point cloud registration

“…The eigenvalue of Y is non-negative. According to SDRSAC [54] and DS++ [24], the optimization of correspondences is to solve the problem of max XY AY with four conditions: (1) X should be {0, 1}, (2) the row sum of X should be no larger than 1, (3) the column sum of X should be no larger than 1, and (4) the sum of X should equal to the number of correspondence pairs. By solving the above maximization problem, we can obtain the global solution of correspondences.…”

“…The registration problem has endured thorough investigation from optimization aspects [5], [6], [24], [33], [44], [47], [54], [90], [104]. Most of the existing registration methods are formulated by minimizing a geometric projection error through two processes: correspondence searching and transformation estimation.…”

A Dual Iterative Refinement Method for Non-rigid Shape Matching

“…Although the matching of pairwise descriptors is stricter and may be more robust and accurate, it requires to solve a quadratic assignment problem (QAP) which is NP-hard. Various methods have been proposed to solve the QAP approximately in a more computational tractable way e.g sub-sampling [49], coarse-to-fine [57], geodesic distance sparsity enforcement methods [19] and various relaxation approaches [1,8,25,30,13,17]. One popular approach is to relax the nonconvex permutation matrix (representing pointwise correspondence) constraint in the QAP to a doubly stochastic matrix (convex) constraint [1,13].…”