基于矩阵平衡的点集匹配内点方法
Matrix balancing based interior point methods for point set matching problems
摘要：
点集匹配问题可以用最优输运来处理。其背后的机制是，最优输运恢复与最小卷曲变形相关联的点对点对应关系。最优输运是一种特殊形式的密集约束线性规划。线性规划可以用内点方法来处理，只要所涉及的病态Hessian能被精确计算。在这十年中，矩阵平衡已被用于计算熵正则化方法下的最优输运。内点法的解的质量取决于两个因素：矩阵平衡的精度和对偶向量的有界性。为了实现高精度的矩阵平衡，我们采用牛顿法实现沿一条中心路径的一系列矩阵的矩阵平衡，在这项工作中，我们将稀疏支持约束应用于基于矩阵平衡的内点方法，其中满足总支持的稀疏集被迭代更新以截断运输计划的域。全支撑条件是保证矩阵平衡存在和对偶向量有界的一个重要条件。
英文摘要：
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.
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