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Expected Sliced Transport Plans
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Proposes Expected Sliced Transport (EST) plans, built by lifting optimal 1-D plans from many projections and averaging them, and proves the induced discrepancy is a metric on discrete measures that connects to min-SWGG via a temperature parameter.
Problem
Sliced optimal transport is computationally cheap but yields no transportation plan between the original measures, limiting applications needing explicit coupling; it is unclear whether a plan from sliced transport can define a metric.
Approach
Lift each one-dimensional optimal plan back to the original space (weighting by the masses of projected points), average the lifted plans over slicing directions to get an expected sliced transport plan, and use its cost as a discrepancy; a softmax temperature over slices interpolates between uniform slicing and min-SWGG.
Key results
- EST discrepancy converges to zero along a Wasserstein geodesic toward the target, while entropic OT and outer product plans do not, as shown numerically.
- EST plans with a temperature interpolate between uniform slicing and min-SWGG, with mass splitting decreasing as temperature increases.
- In transport-based embedding on Point Cloud MNIST 2D, the EST embedding becomes less informative as the temperature parameter increases, reported as a function of reference size and temperature.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use the expected sliced transport plan to define a distance between finite discrete probability measures | Expected Sliced Transport (EST) · Finite discrete probability measures | metric validity | Helped |
| Use the expected sliced transport plan instead of the optimal transport plan | Point Cloud MNIST 2D interpolation · Point Cloud MNIST 2D | mass splitting | Mixed |
| Increase temperature parameter τ (ablation) | Expected Sliced Transport plan · Point Cloud MNIST 2D interpolation | mass splitting | Helped |
| Increase temperature parameter τ (ablation) | Expected Sliced Transport plan · Point Cloud MNIST 2D interpolation | computation time | No clear effect |
| Use EST rather than entropic OT or the outer-product plan | Transport plans between μ_t and ν · Weak convergence as μ_t converges to ν | discrepancy | Helped |
| Increase temperature parameter τ when computing EST embeddings (ablation) | EST transport-based embedding · Point Cloud MNIST 2D classification | embedding informativeness | Hurt |
Surprising findings
- Increasing the temperature parameter (sharpening slice weights toward min-SWGG) makes the transport-based embedding progressively less informative for classification.
Limitations
- The embedding built from EST degrades as the temperature grows because slice weights depend on each measure, causing inconsistency across embedded measures.
- Theoretical metric result relies on the optimal 1-D plans and approximation arguments; practical use approximates the expectation with finitely many slices L.
Experimental setup
Datasets: Point Cloud MNIST 2D
Benchmarks: Point Cloud MNIST 2D
Metrics: classification test accuracy (logistic regression on embeddi discrepancy values along Wasserstein geodesic
Baselines: exact optimal transport (W2) entropic optimal transport min-SWGG outer product plan
Builds on: sliced Wasserstein distance min-SWGG linear optimal transport embedding entropic optimal transport
Closest papers by meaning
- Differentiable Generalized Sliced Wasserstein Plans (NeurIPS 2025 · similarity 0.93)
- Amortized Optimal Transport from Sliced Potentials (NeurIPS 2026 · similarity 0.90)
- Efficient Transferable Optimal Transport via Min-Sliced Transport Plans (NeurIPS 2026 · similarity 0.89)
- Low-Rank Optimal Transport through Factor Relaxation with Latent Coupling (NeurIPS 2024 · similarity 0.88)
- Optimal Transport with Tempered Exponential Measures (AAAI 2024 · similarity 0.88)
- Submodular framework for structured-sparse optimal transport (ICML 2024 · similarity 0.87)
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