  Sinkhorn algorithm has been used pervasively to approximate the solution to
optimal transport (OT) and unbalanced optimal transport (UOT) problems.
However, its practical application is limited due to the high computational
complexity. To alleviate the computational burden, we propose a novel
importance sparsification method, called Spar-Sink, to efficiently approximate
entropy-regularized OT and UOT solutions. Specifically, our method employs
natural upper bounds for unknown optimal transport plans to establish effective
sampling probabilities, and constructs a sparse kernel matrix to accelerate
Sinkhorn iterations, reducing the computational cost of each iteration from
$O(n^2)$ to $\widetilde{O}(n)$ for a sample of size $n$. Theoretically, we show
the proposed estimators for the regularized OT and UOT problems are consistent
under mild regularity conditions. Experiments on various synthetic data
demonstrate Spar-Sink outperforms mainstream competitors in terms of both
estimation error and speed. A real-world echocardiogram data analysis shows
Spar-Sink can effectively estimate and visualize cardiac cycles, from which one
can identify heart failure and arrhythmia. To evaluate the numerical accuracy
of cardiac cycle prediction, we consider the task of predicting the end-systole
time point using the end-diastole one. Results show Spar-Sink performs as well
as the classical Sinkhorn algorithm, requiring significantly less computational
time.
