Matrix balancing based interior point methods for point set matching problems

TL;DR

This paper introduces a novel interior point method for point set matching problems using matrix balancing with Newton methods, enhancing accuracy and efficiency in optimal transport computations.

Contribution

It proposes a new approach combining matrix balancing and interior point methods with sparse support constraints for improved optimal transport solutions.

Findings

Achieves high-accuracy matrix balancing using Newton methods.

Ensures bounded dual vectors through total support constraints.

Demonstrates improved performance in point set matching tasks.

Abstract

Point sets matching problems can be handled by optimal transport. The mechanism behind it is that optimal transport recovers the point-to-point correspondence associated with the least curl deformation. Optimal transport is a special form of linear programming with dense constraints. Linear programming can be handled by interior point methods, provided that the involved ill-conditioned Hessians can be computed accurately. During the decade, matrix balancing has been employed to compute optimal transport under entropy regularization approaches. The solution quality in the interior point method relies on two ingredients: the accuracy of matrix balancing and the boundedness of the dual vector. To achieve high accurate matrix balancing, we employ Newton methods to implement matrix balancing of a sequence of matrices along one central path. In this work, we apply sparse support constraints to matrix-balancing based interior point methods, in which the sparse set fulfilling total support is iteratively updated to truncate the domain of the transport plan. Total support condition is one crucial condition, which guarantees the existence of matrix balancing as well as the boundedness of the dual vector.