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Time Parameterized Optimal Transport

Kaiwen Shi

TL;DR

This work extends optimal transport to dynamic settings by introducing time-parameterized OT, enabling sequential decisions over a horizon $N$ under practical constraints. It decomposes the problem into two tractable subproblems: capacity-constrained TOT and sparsity-constrained TOT, and develops both exact and accelerated algorithms. A key contribution is a reformulation that reduces the time-expanded LP into a fixed-dimension problem, yielding substantial computational savings while preserving optimality under certain conditions. To address sparsity, the authors propose a heuristic based on an importance score that prioritizes entries for nonzero transport, trading exact optimality for major runtime gains with competitive solution quality. The results demonstrate scalable efficiency gains and provide a practical framework for dynamic transport planning with capacity and sparsity constraints, alongside insights into the tradeoffs between exactness and speed.

Abstract

Optimal transport has gained significant attention in recent years due to its effectiveness in deep learning and computer vision. Its descendant metric, the Wasserstein distance, has been particularly successful in measuring distribution dissimilarities. While extensive research has focused on optimal transport and its regularized variants (such as entropy, sparsity, and capacity constraints) the role of time has been largely overlooked. However, time is a critical factor in real world transport problems. In this work, we introduce a time parameterized formulation of the optimal transport problem, incorporating a time variable t to represent sequential steps and enforcing specific constraints at each step. We propose a systematic method to solve a special subproblem and develop a heuristic search algorithm that achieves nearly optimal solutions while significantly reducing computational time.

Time Parameterized Optimal Transport

TL;DR

This work extends optimal transport to dynamic settings by introducing time-parameterized OT, enabling sequential decisions over a horizon under practical constraints. It decomposes the problem into two tractable subproblems: capacity-constrained TOT and sparsity-constrained TOT, and develops both exact and accelerated algorithms. A key contribution is a reformulation that reduces the time-expanded LP into a fixed-dimension problem, yielding substantial computational savings while preserving optimality under certain conditions. To address sparsity, the authors propose a heuristic based on an importance score that prioritizes entries for nonzero transport, trading exact optimality for major runtime gains with competitive solution quality. The results demonstrate scalable efficiency gains and provide a practical framework for dynamic transport planning with capacity and sparsity constraints, alongside insights into the tradeoffs between exactness and speed.

Abstract

Optimal transport has gained significant attention in recent years due to its effectiveness in deep learning and computer vision. Its descendant metric, the Wasserstein distance, has been particularly successful in measuring distribution dissimilarities. While extensive research has focused on optimal transport and its regularized variants (such as entropy, sparsity, and capacity constraints) the role of time has been largely overlooked. However, time is a critical factor in real world transport problems. In this work, we introduce a time parameterized formulation of the optimal transport problem, incorporating a time variable t to represent sequential steps and enforcing specific constraints at each step. We propose a systematic method to solve a special subproblem and develop a heuristic search algorithm that achieves nearly optimal solutions while significantly reducing computational time.

Paper Structure

This paper contains 24 sections, 28 equations, 3 figures, 2 tables.

Table of Contents

  1. Background
  2. Introduction
  3. Notation
  4. Time Parameterized Optimal Transport
  5. Time-parameterized Optimal Transport with Capacity Constraints
  6. Problem Statement
  7. A Special Case
  8. An Example of the Special Case and Solution
  9. Observation:
  10. Existence of Solution(s):
  11. A Faster Algorithm to Compute Solution
  12. Observation:
  13. A Faster Algorithm
  14. Experiment
  15. Result
  16. ...and 9 more sections

Figures (3)

  • Figure 1: Visualizaion of pushforwards moving the masses at $x_i$ to position $y_j$peyre2019computational.
  • Figure 2: The sets $B_0$ and $B_1$ as defined in the paper. $B_1$ is the convex hull of $B_0$.
  • Figure 3: Left: LASSO. Right: Ridge. The contour is the level set of the cost function, and the shaded area is the constraint region. It is apparent that it is easier to have a tangent line of the level set going through a corner (a sparse vector) on the left than on the right.