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Towards Better Spherical Sliced-Wasserstein Distance Learning with Data-Adaptive Discriminative Projection Direction
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Proposes Discriminative Spherical Sliced-Wasserstein (DSSW), which reweights projection directions of SSW by a data-adaptive projected energy function (non-parametric or neural-network-based) and evaluates it on gradient flows, earth density estimation, SWAE, and self-supervised learning.
Problem
Spherical Sliced-Wasserstein treats all projection directions equally, which ignores their differing discriminative importance for specific spherical data distributions.
Approach
Weights each projection direction on the Stiefel manifold by a softmax-normalized function of its projected 1D Wasserstein distance, either non-parametrically (exp, identity, poly) or via a learned neural network (linear, nonlinear, attention) over projected supports, yielding a DSSW distance with theoretical guarantees.
Key results
- On earth density estimation, DSSW (linear) achieves best NLL on Earthquake (0.69 vs SSW 0.84) and DSSW (nonlinear) best on Flood (1.20) and Fire (0.05 vs SSW 0.24).
- In mini-batch gradient flows, DSSW (attention) reaches NLL -320.16 vs best baseline ARI-S3W -291.38 and SSW -287.11.
- DSSW (exp) runtime closely tracks original SSW, indicating negligible overhead for the non-parametric variant.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Replace equal-weight SSW directions with data-adaptive DSSW attention weights | Gradient flows on the sphere · Mini-batch | NLL: -291.38 ± 16.48 -> -320.16 ± 5.58 | Helped |
| Replace equal-weight SSW directions with data-adaptive DSSW nonlinear weights | Gradient flows on the sphere · Mini-batch | log W2: -2.82 ± 0.12 -> -2.97 ± 0.15 | Helped |
| Use DSSW with learned linear projection weights instead of ARI-S3W | Earth density estimation · Earthquake | NLL: 0.78 ± 0.06 -> 0.69 ± 0.04 | Helped |
| Use DSSW with learned nonlinear projection weights instead of ARI-S3W | Earth density estimation · Flood | NLL: 1.24 ± 0.04 -> 1.20 ± 0.03 | Helped |
| Use DSSW (exp) instead of ARI-S3W | Earth density estimation · Fire | NLL: 0.10 ± 0.04 -> 0.05 ± 0.08 | Helped |
| Use DSSW with learned attention weights instead of SW | SWAE with vMF prior · CIFAR10 | log W2: -3.2537 ± 0.1116 -> -3.4242 ± 0.0337 | Helped |
| Use DSSW (poly) instead of SW | SWAE with vMF prior · CIFAR10 | NLL: -0.0004 ± 0.0030 -> -0.0018 ± 0.0021 | Helped |
| Use DSSW (linear) instead of SW | SWAE with vMF prior · CIFAR10 | BCE: 0.6307 ± 0.0005 -> 0.6314 ± 0.0004 | Hurt |
| Replace ARI-S3W with DSSW-SSL (linear) | Self-supervised representation learning · CIFAR10, d=10, encoder output | accuracy: 79.92 -> 80.15 | Helped |
| Replace ARI-S3W with DSSW-SSL (linear) | Self-supervised representation learning · CIFAR10, d=10, projected features | accuracy: 75.07 -> 76.87 | Helped |
| Use nonlinear rather than linear learned projection weights (ablation) | DSSW gradient flows · Mini-batch | log W2: -2.94 ± 0.10 -> -2.97 ± 0.15 | Helped |
| Use attention rather than linear learned projection weights (ablation) | DSSW gradient flows · Mini-batch | NLL: -319.41 ± 6.72 -> -320.16 ± 5.58 | Helped |
Surprising findings
- Full-batch gradient flow results show SW achieving the best log W2 (-10.82), outperforming all DSSW variants on that metric.
Limitations
- Parametric projected energy functions add extra training overhead (O(TL)) compared with non-parametric variants.
- DSSW does not satisfy the identity of indiscernibles, only positivity and symmetry.
Experimental setup
Datasets: CIFAR10 MNIST Earthquake Flood Fire von Mises-Fisher mixture (synthetic)
Benchmarks: CIFAR10 Earthquake Flood Fire MNIST
Metrics: NLL log W2 BCE linear evaluation accuracy runtime
Baselines: SW SSW S3W RI-S3W ARI-S3W Stereo Wasserstein distance Sinkhorn distance SimCLR SWAE
Builds on: Spherical Sliced-Wasserstein Discriminative Sliced-Wasserstein Stereographic Spherical Sliced-Wasserstein Sliced-Wasserstein autoencoder Projected gradient descent
Closest papers by meaning
- Stereographic Spherical Sliced Wasserstein Distances (ICML 2024 · similarity 0.91)
- Spherical Sliced-Wasserstein (ICLR 2023 · similarity 0.91)
- Tree-Sliced Wasserstein Distance with Nonlinear Projection (ICML 2025 · similarity 0.88)
- Linear Spherical Sliced Optimal Transport: A Fast Metric for Comparing Spherical Data (ICLR 2025 · similarity 0.86)
- Spherical Tree-Sliced Wasserstein Distance (ICLR 2025 · similarity 0.85)
- Sliced Wasserstein Estimation with Control Variates (ICLR 2024 · similarity 0.85)
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