Expected Sliced Transport Plans

TL;DR

This paper proposes a "lifting"operation to extend one-dimensional optimal transport plans back to the original space of the measures, and proves that using the EST plan to weight the sum of the individual Euclidean costs for moving from one point to another results in a valid metric between the input discrete probability measures.

Abstract

The optimal transport (OT) problem has gained significant traction in modern machine learning for its ability to: (1) provide versatile metrics, such as Wasserstein distances and their variants, and (2) determine optimal couplings between probability measures. To reduce the computational complexity of OT solvers, methods like entropic regularization and sliced optimal transport have been proposed. The sliced OT framework improves efficiency by comparing one-dimensional projections (slices) of high-dimensional distributions. However, despite their computational efficiency, sliced-Wasserstein approaches lack a transportation plan between the input measures, limiting their use in scenarios requiring explicit coupling. In this paper, we address two key questions: Can a transportation plan be constructed between two probability measures using the sliced transport framework? If so, can this plan be used to define a metric between the measures? We propose a "lifting" operation to extend one-dimensional optimal transport plans back to the original space of the measures. By computing the expectation of these lifted plans, we derive a new transportation plan, termed expected sliced transport (EST) plans. We prove that using the EST plan to weight the sum of the individual Euclidean costs for moving from one point to another results in a valid metric between the input discrete probability measures. We demonstrate the connection between our approach and the recently proposed min-SWGG, along with illustrative numerical examples that support our theoretical findings.

Figures (6)

• Figure 1: Visualization of the 1-dimensional plan $\Lambda_\theta^{\mu^1,\mu^2}$ (given an unit vector $\theta$) and the corresponding lifted transport plan $\gamma_\theta^{\mu^1,\mu^2}$ between discrete probability measures $\mu^1$ (green circles) and $\mu^2$ (blue circles). In (a) the measures $\mu^1,\mu^2$ are uniform and the masses do not overlap when projecting in the direction of $\theta$. In (b) the measures $\mu^1,\mu^2$ are not uniform and some of the masses overlap when projecting in the direction of $\theta$. For more details see Remark \ref{['remark-caption']} in Appendix \ref{['app: metric prop prelim']}.
• Figure 2: Depiction of transport plans (an optimal transport plan, a plan obtained from solving an entropically regularized transport problem, and the proposed expected sliced transport plan) between source (orange) and target (blue) for four different configurations of masses. The measures in the left and right panels are concentrated on the same particles, respectively; however, the top row depicts measures with uniform mass, while the bottom row depicts measures with random, non-uniform mass. Transportation plans are shown as gray assignments and as $n\times m$ heat matrices encoding the amount of mass transported (dark color = no transportation, bright color = more transportation), where $n$ is the number of particles on which the source measure is concentrated, and $m=2n$) is the number of particles on which the target measure is concentrated.
• Figure 3: The effect of increasing $\tau$ (i.e., decreasing temperature) on the expected sliced plan. The left most column shows the OT plan, and the rest of the columns show the expected sliced plan as a function of increasing $\tau$. The right most column depicts that expected sliced plan recovers the min-SWGG mahey2023fast transportation map.
• Figure 4: Interpolation between two point clouds via $((1-t)x+ty)_\# \gamma$, where $\gamma$ is the optimal transportation plan for $W_2(\cdot,\cdot)$ (top left), the transportation plan obtained from entropic OT with various regularization parameters (bottom left), and the expected sliced transport plans for different temperatures $\tau$ (right).
• Figure 5: Discrepancies calculated from transportation plans between $\mu_t$ and $\nu$, when $\mu_t\stackrel{*}{\rightharpoonup} \nu$, as a function of $t\in [0,1]$.

Theorems & Definitions (44)

• Remark 2.1
• Remark 2.2
• Lemma 2.3
• Definition 2.4: Expected Sliced Transport plan
• Remark 2.5
• Definition 2.6: Expected Sliced Transport distance
• Remark 2.7
• Remark 2.8
• Remark 2.9: EST for discrete uniform measures and the Projected Wasserstein distance
• Theorem 2.10