Anchor Space Optimal Transport: Accelerating Batch Processing of
  Multiple OT Problems
Jianming Huang Xun Su Zhongxi Fang Hiroyuki Kasai
The optimal transport (OT) theory provides an effective way to compare
probability distributions on a defined metric space, but it suffers from cubic
computational complexity. Although the Sinkhorn's algorithm greatly reduces the
computational complexity of OT solutions, the solutions of multiple OT problems
are still time-consuming and memory-comsuming in practice. However, many works
on the computational acceleration of OT are usually based on the premise of a
single OT problem, ignoring the potential common characteristics of the
distributions in a mini-batch. Therefore, we propose a translated OT problem
designated as the anchor space optimal transport (ASOT) problem, which is
specially designed for batch processing of multiple OT problem solutions. For
the proposed ASOT problem, the distributions will be mapped into a shared
anchor point space, which learns the potential common characteristics and thus
help accelerate OT batch processing. Based on the proposed ASOT, the
Wasserstein distance error to the original OT problem is proven to be bounded
by ground cost errors. Building upon this, we propose three methods to learn an
anchor space minimizing the distance error, each of which has its application
background. Numerical experiments on real-world datasets show that our proposed
methods can greatly reduce computational time while maintaining reasonable
approximation performance.