  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.
