Expected Sliced Plans in Optimal Transport

• Expected Sliced Plans are transport couplings obtained by projecting measures onto one-dimensional slices, solving OT problems, and lifting the solutions back to high dimensions.
• The methodology averages lifted one-dimensional optimal transport plans, offering computational efficiency compared to classical OT while providing an explicit coupling.
• They bridge the gap between cost-intensive classical OT and fast sliced OT, with applications in interpolation, domain adaptation, and attention mechanisms.

Expected Sliced Plans, introduced in the discrete setting as Expected Sliced Transport (EST) plans, are transport couplings obtained by projecting probability measures onto one-dimensional slices, solving the corresponding one-dimensional optimal transport problems, lifting those plans back to the ambient space, and averaging the lifted couplings over directions. They were developed to resolve a structural limitation of sliced Wasserstein methods: sliced distances are computationally efficient, but they ordinarily do not provide an explicit high-dimensional transport plan. EST therefore occupies an intermediate position between classical optimal transport, which yields a bona fide coupling but is expensive, and sliced optimal transport, which is cheap but usually only returns a scalar discrepancy (Liu et al., 2024).

1. Problem setting and motivation

For probability measures $\mu^1,\mu^2 \in \mathcal P_p(\mathbb R^d)$, classical optimal transport defines the $p$-Wasserstein distance by minimizing Euclidean transport cost over couplings $\gamma \in \Gamma(\mu^1,\mu^2)$. In the discrete case,

$$\mu^1=\sum_i p(x_i)\delta_{x_i}, \qquad \mu^2=\sum_j q(y_j)\delta_{y_j},$$

the coupling is a matrix $\gamma_{ij}$ with prescribed marginals, and the resulting linear program has complexity roughly $\mathcal O((\min\{n,m\})^3\log(\min\{n,m\}))$ with standard solvers. Even entropic regularization, while cheaper, still incurs $\mathcal O(nm)$ per Sinkhorn iteration and becomes costly for large supports or small regularization (Liu et al., 2024).

Sliced optimal transport reduces this burden by projecting onto one-dimensional directions $\theta \in \mathbb S^{d-1}$, solving

$W_p\bigl(\theta_\#\mu^1,\theta_\#\mu^2\bigr)$

for each slice, and averaging over directions. In one dimension, optimal transport is obtained by sorting or quantile matching, with complexity around $\mathcal O(n\log n + m\log m)$. The resulting sliced Wasserstein distance is therefore much more scalable, and for empirical measures its Monte Carlo approximation typically costs $p$0 over $p$1 sampled directions. Its principal deficiency is that it averages scalar one-dimensional costs rather than producing a single coupling in $p$2, which limits its use in barycenters, interpolation, domain adaptation, embeddings, and related plan-based tasks (Liu et al., 2024, Tanguy et al., 2 Aug 2025).

2. Discrete Expected Sliced Transport construction

The EST construction begins by solving the unique one-dimensional OT problem on each slice. For a fixed $p$3, the projected measures $p$4 and $p$5 admit a unique optimizer $p$6. The central operation is then a lifting from this one-dimensional plan back to a coupling $p$7 in the original space (Liu et al., 2024).

In the uniform discrete case,

$p$8

the projected points are sorted by permutations $p$9, and the one-dimensional optimizer is the monotone sorted-to-sorted matching. Lifting simply assigns the same index matching in the ambient space: $\gamma \in \Gamma(\mu^1,\mu^2)$0 with matrix representation

$\gamma \in \Gamma(\mu^1,\mu^2)$1

Thus each slice yields a scaled permutation plan (Liu et al., 2024).

For general discrete measures, the lifting must account for projection collisions. Writing $\gamma \in \Gamma(\mu^1,\mu^2)$2 when $\gamma \in \Gamma(\mu^1,\mu^2)$3, the projected atoms are equivalence classes $\gamma \in \Gamma(\mu^1,\mu^2)$4, with class masses

$\gamma \in \Gamma(\mu^1,\mu^2)$5

The lifted weights are then

$\gamma \in \Gamma(\mu^1,\mu^2)$6

whenever $\gamma \in \Gamma(\mu^1,\mu^2)$7 and $\gamma \in \Gamma(\mu^1,\mu^2)$8. This proportional splitting redistributes the mass assigned by the one-dimensional plan across all pairs within the corresponding source and target projection fibers. Lemma 2.1 shows that the resulting $\gamma \in \Gamma(\mu^1,\mu^2)$9 indeed has marginals $$\mu^1=\sum_i p(x_i)\delta_{x_i}, \qquad \mu^2=\sum_j q(y_j)\delta_{y_j},$$0 and $$\mu^1=\sum_i p(x_i)\delta_{x_i}, \qquad \mu^2=\sum_j q(y_j)\delta_{y_j},$$1 (Liu et al., 2024).

The Expected Sliced Transport plan is the directional expectation

$$\mu^1=\sum_i p(x_i)\delta_{x_i}, \qquad \mu^2=\sum_j q(y_j)\delta_{y_j},$$2

where $$\mu^1=\sum_i p(x_i)\delta_{x_i}, \qquad \mu^2=\sum_j q(y_j)\delta_{y_j},$$3 is a probability measure on $$\mu^1=\sum_i p(x_i)\delta_{x_i}, \qquad \mu^2=\sum_j q(y_j)\delta_{y_j},$$4, typically uniform or absolutely continuous with respect to the uniform measure. In practice,

$$\mu^1=\sum_i p(x_i)\delta_{x_i}, \qquad \mu^2=\sum_j q(y_j)\delta_{y_j},$$5

for Monte Carlo directions $$\mu^1=\sum_i p(x_i)\delta_{x_i}, \qquad \mu^2=\sum_j q(y_j)\delta_{y_j},$$6 (Liu et al., 2024).

3. Induced discrepancies, metric structure, and generic-measure extensions

Given the EST plan, the original paper defines the Expected Sliced Transport distance

$$\mu^1=\sum_i p(x_i)\delta_{x_i}, \qquad \mu^2=\sum_j q(y_j)\delta_{y_j},$$7

Its main theorem states that $$\mu^1=\sum_i p(x_i)\delta_{x_i}, \qquad \mu^2=\sum_j q(y_j)\delta_{y_j},$$8 is a metric on the space of finite discrete probability measures in $$\mu^1=\sum_i p(x_i)\delta_{x_i}, \qquad \mu^2=\sum_j q(y_j)\delta_{y_j},$$9. The proof combines symmetry of the slicewise construction, the lower bound $\gamma_{ij}$0, an argument through uniform discrete measures where the quantity coincides with the Projected Wasserstein distance of Rowland et al. (2019), an extension to rational weights by atom replication, and an approximation argument for general finite discrete measures. Under compactness and $\gamma_{ij}$1, the same work also shows that $\gamma_{ij}$2 induces the same topology as $\gamma_{ij}$3 on discrete measures (Liu et al., 2024).

A later generalization reformulates Expected Sliced Plans for generic measures in $\gamma_{ij}$4 by disintegrating each $\gamma_{ij}$5 along the projection map $\gamma_{ij}$6: $\gamma_{ij}$7 For a one-dimensional optimal plan $\gamma_{ij}$8 between $\gamma_{ij}$9 and $\mathcal O((\min\{n,m\})^3\log(\min\{n,m\}))$0, the lifted plan $\mathcal O((\min\{n,m\})^3\log(\min\{n,m\}))$1 couples the fibers $\mathcal O((\min\{n,m\})^3\log(\min\{n,m\}))$2 and $\mathcal O((\min\{n,m\})^3\log(\min\{n,m\}))$3 independently conditional on $\mathcal O((\min\{n,m\})^3\log(\min\{n,m\}))$4. Averaging over $\mathcal O((\min\{n,m\})^3\log(\min\{n,m\}))$5 produces

$\mathcal O((\min\{n,m\})^3\log(\min\{n,m\}))$6

and the associated Expected Sliced discrepancy is

$\mathcal O((\min\{n,m\})^3\log(\min\{n,m\}))$7

This generic-measure analysis shows that the construction remains canonical and measurable, but it also clarifies an important limitation: $\mathcal O((\min\{n,m\})^3\log(\min\{n,m\}))$8 is not a true metric on all of $\mathcal O((\min\{n,m\})^3\log(\min\{n,m\}))$9, because self-distance can be positive for continuous measures. Two explicit counterexamples are given. For $\mathcal O(nm)$0 and $\mathcal O(nm)$1, one gets $\mathcal O(nm)$2. For $\mathcal O(nm)$3 uniform on the unit ball in $\mathcal O(nm)$4, one gets $\mathcal O(nm)$5. By contrast, if $\mathcal O(nm)$6 is absolutely continuous with respect to the uniform measure and $\mathcal O(nm)$7 is countably discrete, then projection collisions occur only on a null set of directions, and $\mathcal O(nm)$8 becomes a true distance on the countably discrete class $\mathcal O(nm)$9 (Tanguy et al., 2 Aug 2025).

4. Position within sliced-plan methodology

Expected Sliced Plans belong to a broader family of sliced constructions that attempt to recover couplings, not merely distances. For uniform discrete measures, the EST distance can be written as

$\theta \in \mathbb S^{d-1}$0

which coincides with the Projected Wasserstein distance studied by Rowland et al. (2019). The same paper also introduces a temperature scheme

$\theta \in \mathbb S^{d-1}$1

that interpolates between uniform averaging over directions at $\theta \in \mathbb S^{d-1}$2 and concentration on the minimizing slice as $\theta \in \mathbb S^{d-1}$3, thereby connecting EST to min-SWGG (Liu et al., 2024).

Later work on Differentiable Generalized Sliced Wasserstein Plans adopts the min-SWGG philosophy rather than global averaging. It formulates min-GSWP as a bilevel problem: the inner problem solves one-dimensional OT after a scalar feature map $\theta \in \mathbb S^{d-1}$4, while the outer problem evaluates the lifted plan in the original cost. Because the outer value function is piecewise constant or discontinuous in $\theta \in \mathbb S^{d-1}$5, that work introduces a smoothed plan

$\theta \in \mathbb S^{d-1}$6

based on Gaussian perturbations and Stein’s lemma. This is literally an expected sliced plan, but only locally around a candidate slice, and its purpose is differentiability of the outer optimization rather than averaging over a global directional law (Chapel et al., 28 May 2025).

The 2025 analysis of generic sliced plans contrasts Expected Sliced Plans with Pivot Sliced Discrepancy. Pivot Sliced Discrepancy is derived from the $\theta \in \mathbb S^{d-1}$7-based Wasserstein distance, admits an exact constrained Kantorovich formulation, and is shown to be a semi-metric on $\theta \in \mathbb S^{d-1}$8 and a metric under additional assumptions on projections. Expected Sliced Plans, by contrast, are obtained by independent lifting along orthogonal fibers, typically yield dense probabilistic couplings, and do not admit an analogous constrained optimization interpretation (Tanguy et al., 2 Aug 2025).

A concise comparison is therefore:

Construction	Slice usage	Plan character
Expected Sliced / EST	Average lifted plans over many directions	Explicit averaged coupling, typically dense
min-SWGG / DGSWP	Select or optimize a single slice	Explicit single-slice coupling, upper-bounding OT
Pivot Sliced Discrepancy	Enforce a constrained projected coupling	Constrained Wasserstein plan, often sparse
Sliced Wasserstein distance	Average one-dimensional costs only	No global coupling

Taken together, these works suggest a three-way distinction between averaging slices, optimizing a slice, and constraining a slice; the main technical differences concern admissibility, metric properties, sparsity, and the extent to which the resulting coupling approximates a full OT plan.

5. Computational realizations and differentiable variants

For empirical measures $\theta \in \mathbb S^{d-1}$9 and $W_p\bigl(\theta_\#\mu^1,\theta_\#\mu^2\bigr)$0, a practical EST pipeline consists of sampling $W_p\bigl(\theta_\#\mu^1,\theta_\#\mu^2\bigr)$1 directions, projecting points, aggregating overlapping projections into one-dimensional atoms, solving one-dimensional OT by sorting and cumulative-mass matching, lifting each one-dimensional optimizer with the proportional splitting rule, averaging the lifted plans, and finally evaluating the Euclidean cost under the averaged coupling. Per slice, the cost is $W_p\bigl(\theta_\#\mu^1,\theta_\#\mu^2\bigr)$2, and the total complexity is

$W_p\bigl(\theta_\#\mu^1,\theta_\#\mu^2\bigr)$3

with the map $W_p\bigl(\theta_\#\mu^1,\theta_\#\mu^2\bigr)$4 piecewise constant for finite supports (Liu et al., 2024).

The generic-measure formulation preserves essentially the same Monte Carlo structure. For empirical measures, one computes the projected matrix $W_p\bigl(\theta_\#\mu^1,\theta_\#\mu^2\bigr)$5 of the one-dimensional optimal plan, the normalization factors

$W_p\bigl(\theta_\#\mu^1,\theta_\#\mu^2\bigr)$6

forms the lifted coupling

$W_p\bigl(\theta_\#\mu^1,\theta_\#\mu^2\bigr)$7

and averages over directions. This retains the $W_p\bigl(\theta_\#\mu^1,\theta_\#\mu^2\bigr)$8 scaling that makes sliced couplings attractive when full OT is infeasible (Tanguy et al., 2 Aug 2025).

A distinct computational development appears in ESPFormer, which uses Expected Sliced Transport Plans as an attention mechanism. In the uniform equal-support case, each slice $W_p\bigl(\theta_\#\mu^1,\theta_\#\mu^2\bigr)$9 is implemented through differentiable soft sorting: $\mathcal O(n\log n + m\log m)$0 followed by

$\mathcal O(n\log n + m\log m)$1

Axis-aligned slices are used, with $\mathcal O(n\log n + m\log m)$2, so every feature dimension acts as a slicer. Slice weights are then set by an inverse-temperature softmax,

$\mathcal O(n\log n + m\log m)$3

and the final attention matrix is

$\mathcal O(n\log n + m\log m)$4

Because each $\mathcal O(n\log n + m\log m)$5 is a scaled permutation in the hard-sorting limit and a row/column-balanced matrix under SoftSort, $\mathcal O(n\log n + m\log m)$6 is approximately doubly-stochastic, and exactly doubly-stochastic as $\mathcal O(n\log n + m\log m)$7. This construction avoids iterative Sinkhorn normalization, is fully parallelizable across slices, and has overall complexity $\mathcal O(n\log n + m\log m)$8, compared with $\mathcal O(n\log n + m\log m)$9 for Sinkformer with $p$00 Sinkhorn iterations (Shahbazi et al., 11 Feb 2025).

6. Empirical behavior, applications, and limitations

The original EST experiments compare classical OT, entropic OT, and EST on simple two-dimensional point clouds. OT plans are sparse and typically close to permutations; entropic plans are much more diffuse; EST lies between them, displaying some mass splitting but generally less than entropic OT. On Point Cloud MNIST 2D, EST-based interpolations behave well visually, and increasing the temperature parameter $p$01 sharpens the interpolation by reducing splitting. In a weak-convergence experiment along a Wasserstein geodesic $p$02, only OT and EST discrepancies go to $p$03 as $p$04; entropic OT and the outer product $p$05 do not. Within a linear optimal transport embedding pipeline, EST embeddings are competitive with OT-based embeddings for moderate $p$06, but performance degrades as $p$07 increases because different samples then use different effective slicing measures $p$08, reducing comparability across embeddings (Liu et al., 2024).

ESPFormer repurposes the same structural idea for attention regularization. Across image classification, point cloud classification, sentiment analysis, and neural machine translation, the reported experiments show consistent gains over vanilla attention and Sinkhorn-based attention. On IWSLT’14 De–En, after 10 epochs of fine-tuning, ESPFormer achieves BLEU scores of $p$09 and $p$10 in the two tested settings. The paper further reports that gains are especially pronounced in low-data or fine-grained regimes, and attributes this to the doubly-stochastic, more balanced attention patterns induced by ESP (Shahbazi et al., 11 Feb 2025).

The broader sliced-plan literature also exposes important limitations of expected averaging. In gradient-flow experiments, Expected Sliced discrepancy often does not converge to the target, especially in higher dimension or complex geometries. In color transfer, barycentric projections of Expected Sliced Plans often produce duller, less faithful colors than alternative sliced-plan methods. In shape registration, Expected Sliced remains usable as a source of soft correspondences, but Min-Pivot Sliced generally yields better registration quality. By contrast, DGSWP, which optimizes a single generalized slice rather than averaging many of them, performs well for gradient flows in Euclidean and hyperbolic spaces and for conditional flow matching in image generation, where it yields better FID than OT-CFM and I-CFM in the 100-step Euler sampling regime (Tanguy et al., 2 Aug 2025, Chapel et al., 28 May 2025).

Several limitations recur across these works. EST is a non-optimal plan, so its induced cost satisfies $p$11. Approximation quality depends on the number of slices $p$12, and convergence rates in $p$13 are not fully analyzed in the original paper. High temperatures in the min-SWGG interpolation scheme can harm transport-based embeddings because the direction distribution becomes pair-dependent. The generic-measure theory shows that positive self-distance can occur for continuous measures, so metricity is restricted to discrete settings unless the construction is modified. These observations support a more precise view of Expected Sliced Plans: they are principled, explicit, and computationally tractable couplings that successfully bridge sliced OT and plan-based transport in the discrete regime, but their theoretical behavior and empirical suitability depend strongly on how directions are averaged, how lifting is performed, and whether the target application benefits from soft averaged couplings or from sharper single-slice approximations (Liu et al., 2024, Tanguy et al., 2 Aug 2025).