Safe Screening for Unbalanced Optimal Transport
This work addresses computational bottlenecks in optimal transport for applications like machine learning and data science, though it appears incremental as it extends existing Safe Screening methods to a specific problem variant.
The paper tackles the computational challenge of Unbalanced Optimal Transport (UOT) by applying Safe Screening techniques to identify and eliminate zero elements in sparse solutions, accelerating optimization without increasing algorithm complexity. They demonstrate feasibility for UOT with ℓ₂ and KL penalties through theoretical analysis and propose novel methods like approximate projection and elliptical safe regions, achieving significant efficiency improvements.
This paper introduces a framework that utilizes the Safe Screening technique to accelerate the optimization process of the Unbalanced Optimal Transport (UOT) problem by proactively identifying and eliminating zero elements in the sparse solutions. We demonstrate the feasibility of applying Safe Screening to the UOT problem with $\ell_2$-penalty and KL-penalty by conducting an analysis of the solution's bounds and considering the local strong convexity of the dual problem. Considering the specific structural characteristics of the UOT in comparison to general Lasso problems on the index matrix, we specifically propose a novel approximate projection, an elliptical safe region construction, and a two-hyperplane relaxation method. These enhancements significantly improve the screening efficiency for the UOT's without altering the algorithm's complexity.