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Sliced Wasserstein with Random-Path Projecting Directions
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Proposes random-path projecting directions (normalized differences of samples from two measures, smoothed by a vMF/power-spherical perturbation) to build an optimization-free slicing distribution, yielding RPSW and IWRPSW sliced Wasserstein variants evaluated on gradient flows and diffusion GAN training.
Problem
Slicing distribution selection for sliced Wasserstein: existing discriminative choices (DSW, Max-SW) need expensive optimization, while energy-based slicing (EBSW) needs importance sampling or MCMC with biased estimates.
Approach
Define the projecting direction as the normalized difference of random samples from the two input measures, perturbed by a location-scale distribution on the sphere (vMF or power spherical). This gives an intractable but i.i.d.-sampleable slicing distribution (RPSD) that is continuous and discriminative with no optimization; IWRPSW adds energy-based importance weights over L directions.
Key results
- On CIFAR10 with 4-step implicit denoising diffusion, IWRPSW-DD reaches FID 2.70 versus 3.64 for DDGAN and 2.82 for RPSW-DD.
- In toy gradient flows, RPSW and IWRPSW reach lower W2 than SW and EBSW at comparable runtime (e.g., IWRPSW 21.81 vs SW 25.17 at step 5000).
- Gradient flow on MNIST: RPSW and IWRPSW reduce W2 faster than SW and EBSW, with much lower runtime than DSW and Max-SW.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Replace the DDGAN objective with AGME using an RPSW kernel | Implicit denoising model on CIFAR10 · CIFAR10 | FID: 3.64 -> 2.82 | Helped |
| Use an IWRPSW kernel instead of an RPSW kernel | Implicit denoising model on CIFAR10 · CIFAR10 | FID: 2.82 -> 2.70 | Helped |
| Use RPSW instead of SW | Toy gradient flow · Toy gradient flow, Step 5000 | W2: 25.17 -> 21.86 | Helped |
| Use RPSW instead of SW | Toy gradient flow · Toy gradient flow, Step 5000 | Time (s): 107.83 -> 118.83 | Hurt |
| Use IWRPSW instead of EBSW | Toy gradient flow · Toy gradient flow, Step 5000 | W2: 25.11 -> 21.81 | Helped |
| Use the original gradient estimator instead of the simplified gradient estimator (ablation) | RPSW and IWRPSW gradient flows · Toy gradient flow | Convergence speed and source-topology preservation | Mixed |
Surprising findings
- The paper reports that the simplified gradient estimator (dropping phi-dependence in the slicing distribution) converges faster but preserves source topology worse than the original estimator.
Limitations
- Triangle inequality for RPSW and IWRPSW is unproven, so they are only semi-metrics and two-sided sample complexity is not established.
- The random path is a straight line, so extension to manifolds such as the sphere or hyperbolic space needs different definitions.
Experimental setup
Datasets: CIFAR10 MNIST
Benchmarks: CIFAR10 MNIST
Metrics: FID Wasserstein-2 distance computational time NFE
Baselines: SW Max-SW DSW EBSW DDGAN DDPM Score SDE LSGM
Builds on: sliced Wasserstein distributional sliced Wasserstein energy-based sliced Wasserstein denoising diffusion GAN mini-batch energy distance
Closest papers by meaning
- Revisiting Sliced Wasserstein on Images: From Vectorization to Convolution (NeurIPS 2022 · similarity 0.90)
- Markovian Sliced Wasserstein Distances: Beyond Independent Projections (NeurIPS 2023 · similarity 0.90)
- Quasi-Monte Carlo for 3D Sliced Wasserstein (ICLR 2024 · similarity 0.89)
- Sliced Wasserstein Estimation with Control Variates (ICLR 2024 · similarity 0.89)
- Assessing the quality of denoising diffusion models in Wasserstein distance: noisy score and optimal bounds (NeurIPS 2025 · similarity 0.86)
- Hierarchical Hybrid Sliced Wasserstein: A Scalable Metric for Heterogeneous Joint Distributions (NeurIPS 2024 · similarity 0.86)
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