Stabilized Sparse Scaling Algorithms for Entropy Regularized Transport Problems
For practitioners of optimal transport, this work provides a more efficient and stable algorithm for large-scale entropic transport problems, though it is an incremental improvement combining existing techniques.
The paper addresses numerical limitations of scaling algorithms for entropy-regularized transport problems, such as divergence of scaling factors and slow convergence near zero regularization. By combining log-domain stabilization, epsilon-scaling, adaptive kernel truncation, and coarse-to-fine schemes, the method solves larger problems with smaller regularization and negligible truncation error.
Scaling algorithms for entropic transport-type problems have become a very popular numerical method, encompassing Wasserstein barycenters, multi-marginal problems, gradient flows and unbalanced transport. However, a standard implementation of the scaling algorithm has several numerical limitations: the scaling factors diverge and convergence becomes impractically slow as the entropy regularization approaches zero. Moreover, handling the dense kernel matrix becomes unfeasible for large problems. To address this, we combine several modifications: A log-domain stabilized formulation, the well-known epsilon-scaling heuristic, an adaptive truncation of the kernel and a coarse-to-fine scheme. This permits the solution of larger problems with smaller regularization and negligible truncation error. A new convergence analysis of the Sinkhorn algorithm is developed, working towards a better understanding of epsilon-scaling. Numerical examples illustrate efficiency and versatility of the modified algorithm.