Neural Projection Fusion for Sliced Wasserstein on the Hypersphere

1 Introduction

In many real-world applications, data often resides on hypersphere, highlighting the critical role of spherical geometry across a wide range of tasks. Typical examples include geophysical and meteorological modeling [Di Marzio et al., 2014; Besombes et al., 2021], magnetic brain imaging in medical analysis [Vrba and Robinson, 2001; Hu et al., 2023; Hu et al., 2025], and texture mapping in computer graphics [Dominitz and Tannenbaum, 2009]. In such settings, latent representations are usually constrained to bounded domains that are well-approximated by unit spheres [Wang and Isola, 2020; Chen et al., 2020]. These settings call for learning methods designed specifically for spherical manifolds.

1∗Corresponding Author.

Traditional analysis on the hypersphere falls within spherical statistics [Jammalamadaka and Sengupta, 2001], which studies directional data such as orientations and rotations. Recently, Optimal Transport (OT) theory has gained attention for comparing distributions on the hypersphere [Cui et al., 2019], due to its strong statistical and geometric properties [Peyr´e et al., 2019]. However, OT often suffer from high computational complexity and the curse of dimensionality [Peyr´e and Cuturi, 2019], which has motivated the development of scalable solvers [Cuturi, 2013] and computationally efficient alternative distance metrics [Kolouri et al., 2019a]. Among them, the Sliced-Wasserstein (SW) [Nguyen et al., 2021; Ohana et al., 2023] distance preserves the topological properties of the Wasserstein distance [Nadjahi et al., 2020], while being more efficient and immune to the curse of dimensionality [Nietert et al., 2022].

The SW distance has recently been extended to spherical domains for distribution comparison, which has led to several spherical sliced OT approaches [Bonet et al., 2023; Quellmalz et al., 2023; Tran et al., 2024]. A key challenge is to generalize the classical Radon transform to the sphere. [Quellmalz et al., 2023] introduced two such extensions, the vertical slice transform [Rubin, 2018] and the normalized semicircle transform [Groemer, 1998]. Bonet et al. [Bonet et al., 2023] proposed a new spherical Radon transform and used the closed-form 1D Wasserstein distance on the circle to define Spherical Sliced-Wasserstein (SSW). Although both works [Bonet et al., 2023; Quellmalz et al., 2023] project spherical distributions onto circular marginals, computing OT on the circle remains relatively expensive. To reduce the computation cost, Tran et al. [Tran et al., 2024] mapped the hypersphere to a hyperplane via stereographic projection and then applied the classical Radon transform to define Stereographic Sliced-Wasserstein (S3W). Tran et al. [Tran et al., 2025] further proposed the Spherical Tree-Sliced Wasserstein (STSW) distance, which improves scalability by combining the spherical Radon transform with tree-based metrics.

Most existing spherical sliced Wasserstein methods sample projection directions uniformly, implicitly treating all directions as equally informative. However, in high-dimensional spherical spaces, discriminative information is often concentrated in only a few directions, making many random projections weakly informative. Recent adaptive methods such as EBSW [Nguyen and Ho, 2024] and DSSW [Zhang et al., 2025] reweight projections according to their onedimensional discrepancies, but still rely on fixed sampled directions and cannot correct poorly aligned projections. MaxSW [Deshpande et al., 2019] selects the most discriminative projection, but sacrifices the averaging structure of sliced distances and may lead to unstable optimization.

To address these limitations, we propose Fusion Stereographic Sliced Wasserstein (FS3W). Rather than reweighting or selecting fixed projections, FS3W learns to refine projection directions themselves. Starting from uniformly sampleddirections, a neural slicing fusion mechanism generates datadependent directions jointly from the two measures, better capturing their geometric structure while preserving the integral form of sliced Wasserstein distances. Thus, FS3W retains the stability and interpretability of averaging-based formulations while improving expressiveness through principled data adaptivity. Our contributions are as follows:

• We propose a neural slicing approach that adaptively refines projection directions for more informative and discriminative hyperspherical comparisons.
• We provide comprehensive theoretical analysis to guarantee the topological and statistical properties of FS3W.
• We validate the effectiveness of FS3W on several classical machine learning tasks, showing that it consistently achieves superior accuracy compared to SSW, S3W, STSW, SW, and the Wasserstein distance.

2 Background

This section reviews the Wasserstein distance, sliced Wasserstein (SW), and stereographic spherical sliced Wasserstein (S3W), which form the foundation of our proposed FS3W on the hypersphere Sd = {s ∈ Rd+1 | ∥s∥2 = 1}.

Wasserstein Distance. Let M be a Riemannian manifold with distance c(·, ·). For p ≥ 1, denote by Pp(M) the set of probability measures with finite p-th moment. The p-Wasserstein distance between µ, ν ∈ Pp(M) is defined as [Peyr´e and Cuturi, 2019]

where Π(µ, ν) denotes the set of couplings between µ and ν. While Wasserstein distance has strong theoretical properties, its computation is costly for empirical measures, which motivates more efficient alternatives.

Sliced Wasserstein Distance. For one dimensional measures µ, ν ∈ Pp(Rd ), the Wasserstein distance between µ and ν admits the closed form as:

where F -1 µ and F -1 ν are quantile functions. This leads to the sliced Wasserstein (SW) distance [Bonneel et al., 2015],

where σd is the uniform distribution on the unit sphere Sd-1 . The Radon Transform [Helgason and others, 2011]

R : L1 (Rd ) → L1 (R×Sd-1 ) is defined as: Rf (t, θ) = ∫Rd f (x) · δ (t - ⟨x, θ⟩) dx, where L1 is the set of absolutely integrable functions on Rd and δ denotes the Dirac delta function. Since the expectation in Eq. (3) is intractable, it is approximated via Monte Carlo estimation with computational complexity O(Ln(d+ logn)), where n and d denote the sample size and data dimensionality, respectively:

where the directions {θℓ}Lℓ=1 are sampled independently and uniformly from U(Sd-1 ), and L is the number of projections. Stereographic Projection. Stereographic projection is an explicit map that transforms points on a sphere onto a plane. The stereographic projection ϕ : Sd \ {sn} → Rd maps a point s ∈ Sd (excluding the "North Pole," sn = [0, . . . , 0, 1]) to a point x ∈ Rd as [Tran et al., 2024]:

This projection is smooth and bijective between Sd \{sn} and Rd , corresponding to the hyperplane tangent to the sphere at the south pole (sd+1 = -1) and commonly used to visualize spherical geometry in Euclidean space.

Stereographic Spherical Radon Transform. Let µ ∈ M(Sd ) be a Radon measure defined on Sd that does not assign mass to the North Pole, i.e., µ({sn}) = 0. Consider the stereographic projection ϕ : Sd \ {sn} → Rd , which is bijection, so that ϕ#µ defines a Radon measure on Rd . Based on this mapping, the stereographic spherical Radon transform [Tran et al., 2024] is defined as:

Stereographic Spherical Sliced Wasserstein. Let µ, ν ∈ Pp(Sd ) denote two probability measures on the unit sphere in Rd+1 . Based on the stereographic spherical Radon transform, the stereographic spherical sliced Wasserstein (S3W) distance [Tran et al., 2024] is defined as:

To maintain the maintain rotational symmetry of spherical OT, Tran et al. [Tran et al., 2024] introduce two rotationally invariant extensions of S3W, termed RI-S3W and ARI-S3W. RI-S3W attains invariance by integrating over the sphere's rotation group, while ARI-S3W offers a more efficient approximation via sampling a finite set of rotations.

3 Method

The Spherical Sliced Wasserstein distance compares hyperspherical distributions by averaging one-dimensional Wasserstein distances over uniformly sampled projection directions. However, uniformly sampled directions may differ in their discriminative power, especially in high-dimensional spaces.

To address this, we propose a neural slicing fusion mechanism that injects data-driven information into projection selection. Rather than using fixed directions, our method adapts projections to the input distributions and aligns them with discriminative geometric structures. This yields FS3W, which preserves the integral form of Sliced Wasserstein distances while offering a more expressive and adaptive metric for hyperspherical probability measures.

Definition 1 (Neural Slicing Fusion Mechanism). Let µ, ν ∈ Pp(Sd ) denote two probability measures on the unit sphere in Rd+1 , given a neural network hψ : R(2n+1)∗d → R1∗d , a neural slicing fusion mechanism aψ : Rn∗d ×Rn∗d × R1∗d → R1∗d is defined as:

The neural network hψ can be implemented with a standard nonlinear architecture or a linear self-attention mechanism with learnable parameters ψ. This mechanism generates datadependent projection directions, which are used to compare the distributions. Further details on the architecture of hψ are provided in Supplementary Material Section B.1.

Definition 2 (FS3W). For p ≥ 1, let µ, ν ∈ Pp(Sd ) denote two probability measures on the unit sphere in Rd+1 , and let aψ(µ, ν, θ) be the neural slicing fusion mechanism. We define the FS3W distance as:

where σd is the uniform distribution on Sd-1 , W p p is the pWasserstein distance, and SR(·)θ denotes the stereographic spherical Radon transform. For simplicity, we use FS3W2 to denote FS3WR,2 in the subsequent experiments.

Monte Carlo Estimation. Similar to the S3Wdistance, we adopt Monte-Carlo estimation to approximate FS3W. We randomly sample projection directions θ1, . . . , θL i.i.d ∼ σd ∈ U(Sd-1 ). The estimator of FS3W is then given by:

To ensure projections capture the most discriminative geometric structures, we maximize the discrepancy between projected measures. Specifically, the training objective (MaxFS3W) for mechanism aψ is:

where L is the number of projection directions.

We employ AdamW with a learning rate of 1.0e-3 and 50 iterations for all tasks. The implementation pseudocode of our FS3W variant, RI-FS3W variant, and ARI-FS3W variant is presented in Algorithms 1, 2, and 3 (Supplementary Material Section B.2), respectively. During training, hψ learns to integrate data-specific distribution information into prior projection directions, improving the ability of FS3W to measure distributional discrepancies more accurately.

To implement the fusion mechanism aψ, we employ a nonlinear neural network and linear self-attention [Nguyen et al., 2023], yielding two FS3W variants: NFS3W and LAFS3W. Extending RI-S3W and ARI-S3W with these architectures produces four rotationally invariant variants: RI-NFS3W, RILAFS3W, ARI-NFS3W, and ARI-LAFS3W.

Proposition 1 (Topological properties). Let p ≥ 1 and let aψ(µ, ν, θ) be a neural slicing fusion mechanism satisfying aψ(µ, ν, θ) = aψ(ν, µ, θ). Then the functional FS3WR,p is non-negative and symmetric.

Proposition 1 shows that FS3W preserves the basic topological properties: non-negativity and symmetry. These properties come directly from the corresponding properties of the underlying S3W distance and the symmetry of the fusion mechanism. All proofs for the propositions and theories in this paper are detailed in Supplementary Material Section A.

Proposition 2 (Convergence property). For any p ≥ 1 and the neural slicing fusion mechanism aψ(µ, ν, θ), assume that the stereographic projection is well-defined on the support of µ and that aψ is continuous with respect to (µ, ν, θ). Let µk, µ ∈ Pp(Sd ). If µk ⇒ µ and supk ∫Sd ∥s∥p dµk(s) < ∞, then limk→∞ FS3W pR,p (µk, µ; aψ) = 0.

Proposition 2 establishes the convergence of FS3W under weak convergence of probability measures with bounded moments. This result ensures that FS3W is consistent with standard notions of convergence used in optimal transport and defines a valid discrepancy between probability measures.

Proposition 3 (Sample complexity). For any p ≥ 1 and the neural slicing fusion mechanism aψ(µ, ν, θ), assumed to be Lipschitz continuous in θ, let x1, . . . , xn i.i.d. ∼ µ and y1, . . . , yn i.i.d. ∼ ν be observed samples, where µ, ν ∈ P(Sd ) are supported on a compact set. Let ˆµn and ˆ νn be the corresponding empirical measures.Let R be the diameter of the joint support. Then there exists a constant Cp,R > 0 depending only on (p, R) such that

Proposition 3 characterizes the statistical behavior of FS3W estimated from finite samples. The bound reveals that the estimation error decreases with sample size and depends on the problem dimension, consistent with classical sliced distances. This indicates that the neural fusion mechanism does not increase the sample complexity.

Proposition 4 (Projection Complexity). Let µ, ν ∈ Pp(Sd ), and let aψ(µ, ν, θ) be the neural slicing fusion mechanism, assumed to be Lψ-Lipschitz with respect to θ ∈ Sd-1 . Assume that the stereographic Wasserstein functional W p p (·, ·)

is LW-Lipschitz with respect to its projection parameter. Let θ1, . . . , θL i.i.d. ∼ σd-1 be sampled uniformly from the unit sphere Sd-1 . Define the Monte Carlo estimator

Then there exists a constant Cd > 0, depending only on the geometry of Sd-1 , such that

Proposition 4 analyzes the error of Monte Carlo approximation for projection directions. The bound shows the convergence rate depends on the number of projections, problem dimension, and the Lipschitz regularity of the fusion network. This matches existing results for sliced-based methods, confirming FS3W retains their favorable projection complexity.

Proposition 5 (Neural Fusion Deviation Error). Let aψ be a trained neural slicing fusion mechanism and a∗ denote an ideal mapping that produces optimal discriminative projection directions. We define the network-induced error as:

Under mild regularity conditions on the fusion network and stereographic Wasserstein, we have the bound:

Proposition 5 relates the accuracy of FS3W to the quality of the learned fusion mechanism. The bound shows that errors in the learned projection mapping translate linearly into errors in the resulting distance. This result highlights the importance of effective training and provides theoretical support for learning data-adaptive projection directions.

4 Implementation Details

In this section, we present the implementation details and a computational complexity analysis of FS3W, together with a runtime comparison against existing distance metrics.

We follow the standard SW procedure and use two empirical spherical distributions with the same sample size, µ = 1n ∑ni=1 δxi and ν = 1n ∑ni=1 δyi , where each consists of n samples. First, we apply stereographic projection operator ϕ to map the points into the hyperplane, as defined in Eq. (5), then we randomly sample L projections {θℓ}Lℓ=1 from the uniform distribution σ on the unit sphere Sd-1 , thereafter, we adopt the neural slicing fusion mechanism aψ(µ, ν, θ) to obtain the improved projection directions as in Eq. (10). Next, we project the transformed samples onto each projection direction to form the slicing process. Together, the stereographic projection and the slicing process give the stereographic spherical Radon transform in Eq. (6). Finally, we approximate FS3W by Monte Carlo estimation as in Eq. (10).

Computation Complexity. As discussed in S3W [Tran et al., 2024], the stereographic projection serves as a crucial preprocessing step to map samples from the unit hypersphere Sd to the Euclidean space Rd . Performing the stereographic projection for all samples requires a complexity of O(nd), where n is the total number of samples from the source and target distribution and (d+1) is the dimensionality of the data. The slicing process, which projects the transformed samples onto L projection directions, incurs a cost of O(Lnd). Sorting the data projections takes O(Ln logn), and the final distance calculation requires O(Ln). In our FS3W formulation, the neural slicing fusion mechanism introduces an additional cost of O(LnTmax), where Tmax is the maximum iterations for training the nonlinear neural network or linear self-attention mechanism used to generate the enhanced projection directions. Therefore, the overall time complexity of our FS3W is O(Ln(d+ logn+Tmax)). For the RI-FS3W variant, which incorporates NR rotations, the total complexity becomes O(NRLn(d+ logn+Tmax)). In addition, generating NR random rotation matrices requires O (NRd3 ), and applying these rotations to the data incurs a cost of O (NRnd2 ). Consequently, the overall time complexity of RI-FS3W is O (NR (d3 + nd2 + Ln (d + log n + Tmax))).

Runtime Comparison. Following the S3W protocol [Tran et al., 2024], we compare the runtime of different distance metrics by computing distances between a uniform distribution and a von Mises-Fisher distribution on S100 . Results in Fig. 1 are averaged over 50 runs across sample sizes. For all sliced-based methods, we use L = 200 projections. RI-S3W and ARI-S3W use NR = 10 rotations, with an additional pool size of 100 for ARI-S3W. As shown in Fig. 1, FS3W variants are slower than their S3W counterparts due to the extra training cost of neural slicing fusion, yet remain much faster than Wasserstein and comparable to Sinkhorn. Moreover, their runtime grows smoothly with sample size, showing that fusion learning adds only modest overhead while preserving the scalability of sliced-based methods. We further study the effects of dimension, projections, and rotations in Section. C.1, with detailed runtime analysis in Section. C.2.

5 Experiments

In this section, we present experimental results on four core tasks: Gradient Flow, Self-Supervised Learning, Earth Density Estimation, and Sliced-Wasserstein Auto-Encoder. For each task, we report quantitative metrics, qualitative visualizations, and comparisons with a range of state-of-the-art baseline methods. Together, these results show the effectiveness and versatility of our FS3W across different machine learning scenarios. Additional details and results are provided in Supplementary Material Section C.3-C.6. All experiments are implemented in PyTorch [Paszke et al., 2019] on Ubuntu 20.04 using a single NVIDIA A100 GPU with 40GB RAM.

5.1 Gradient Flows on The Sphere

Similarly to S3W [Tran et al., 2024] and STSW [Tran et al., 2025], we use FS3W to learn a target distribution ν from a source distribution µ, initialized as a uniform distribution on the hypersphere, by minimizing FS3W2 (µ, ν). We also employ the Projected Gradient Descent (PGD) algorithm [Madry et al., 2018] to solve this optimization problem. We compare our approach with SW [Bonneel et al., 2015], SSW [Bonet et al., 2023], S3W variants [Tran et al., 2024], DSSW [Zhang et al., 2025] and STSW [Tran et al., 2025].

Following S3W and STSW, we set the target distribution ν as a mixture of 12 vMFs with κ = 50, using 2400 samples evenly allocated across components.

The projected gradient descent [Bonet et al., 2023] can be formulated as follows:

Setup. We set L = 200 trees and k = 5 lines for STSW, and use L = 1000 projections for the other methods. ARIS3W (30) uses 30 rotations from a pool of 1000, while RIS3W (1) and RI-S3W (5) use 1 and 5 rotations, respectively. Batch sizes are 200 and 2400 for mini- and full-batch settings, respectively. We train FS3W variants and other baselines with Adam using lr = 0.01 for full-batch and lr = 0.001 for mini-batch over 500 epochs and 10 runs. Following S3W and STSW, we use log 2-Wasserstein distance and negative loglikelihood (NLL) as evaluation metrics.

Results. The mini-batch results for all methods are reported in Table 1, showing that our FS3W variants outperform the baselines across all metrics. Furthermore, the Molleweide projections of the mini-batch are demonstrated in Fig. 2. We also report the full-batch results in Table A1 and provide the visualization of the Mollweide projections for full-batch projected gradient descent in Fig. C8 (Supplementary Material Section C.3). The full-batch results validate the same conclusion that our FS3W variants achieve better performance in approximating the target distribution than other distances.

The additional sliced-Wasserstein variational inference experiment shows that our FS3W variants consistently outperform or match other state-of-the-art baselines, with full results provided in Supplementary Material Section C.6.

5.2 Earth Density Estimation

We apply FS3W to the density estimation on S2 using an exponential map normalizing flow model [Mathieu and Nickel, 2020]. This model defines an invertible transformation T trained by minimizing minT FS3W2(T#µ, pZ), where µ is the empirical distribution, and pz is a prior distribution on S2 . In this task, we use a uniform prior, following the setup of the earth dataset [Mathieu and Nickel, 2020], which contains Earthquake [NOAA, 2022], Flood [Brakenridge, 2017] and Fire [EOSDIS, 2020]. The density at any point x ∈ S2 is estimated asc fµ (x) = pZ (T (x)) |detJT (x)|, where JT (x) is the Jacobian matrix of T at x.

Following prior work [Bonet et al., 2023; Tran et al., 2024; Tran et al., 2025], we adopt an exponential mapping normalizing flows model with 48 radial blocks and 100 components each, totaling 24000 parameters. The model is then trained with full batch gradient descent using the Adam optimizer.

Setup. We fix L = 1000 projection directions for SW, SSW, S3W, RI-S3W and ARI-S3W, while our FS3W variants only use L = 100 projection directions. We set the learning rate to lr = 0.05 for STSW, S3W, RI-S3W, ARI-S3W and our methods and lr = 0.1 for SW and SSW. We train other sliced dsitances for 20000 epochs, while our methods and STSW are only trained for 10000 epochs.

Results. We compare our method with SW, SSW, S3W variants, DSSW, and STSW, using NLL as the metric. Table 2 shows that ARI-FS3W achieves the best performance, indicating that FS3W models spherical data more effectively than the baselines. Our FS3W variants also require only 100 projection directions, which is fewer than the other methods. Density visualizations for FS3W on test data are provided in Figs. C9-C15 (Supplementary Section C.4) to give a clear view of the model behavior across datasets.

5.3 Sliced-Wasserstein Autoencoder

Similar to S3W [Tran et al., 2024] and STSW [Tran et al., 2025], we integrate FS3W into the Sliced-Wasserstein Auto- Encoder (SWAE) [Kolouri et al., 2019b] framework to evaluate the performance of generative modeling, which forces the latent space distribution to follow a prior distribution ν. let E : X → Sd and D : Sd → X be the parametric encoder and decoder. The training objective of the revised SWAE is

where µ is the source distribution, C (·, ·) is the reconstruction loss implemented as the standard Binary Cross Entropy (BCE) loss, λ means the regularization coefficient.

Setup. Following prior works [Bonet et al., 2023; Tran et al., 2025; Zhang et al., 2025], We train with Adam optimizer with a learning rate of 10-3 and a batch size of 500 over 100 epochs using BCE loss as our reconstruction loss. We fix L = 200 trees and κ = 10 lines for STSW [Tran et al., 2025], we set L = 100 projection directions for all other methods. We use NR = 5 rotations for RI-S3W and ARI-S3W, with ARIS3W using a pool size of 100 random rotations. For CIFAR-

10, use a mixture of 10 vMFs as the prior, setting λ = 1 for STSW, λ = 10 for SSW, λ = 0.01 for our FS3W variants, and λ = 0.001 for SW and S3W variants. In addition, we use the same model architecture from [Tran et al., 2024].

Results. We evaluate FS3W against SW, SSW, S3W variants, DSSW, and STSW on MNIST [LeCun et al., 1998] and CIFAR-10 [Krizhevsky and Hinton, 2009] over S2 . For CIFAR-10, we use a mixture of 10 vMFs as the prior and report log 2-Wasserstein distance, NLL, and BCE loss. As shown in Table 3, FS3W variants achieve the best results across all metrics. Detailed configurations and MNIST results, including FID scores in Table B3 and generated images in Fig. C16, are provided in Supplementary Material Sec. C.5, further confirming the performance gains of FS3W.

5.4 Self-Supervised Representation Learning

Normalizing feature vectors onto the unit hypersphere improves representation quality and helps prevent feature collapse [Wang and Isola, 2020]. As shown in [Wang and Isola, 2020], contrastive learning decomposes into alignment, which pulls positive pairs together, and uniformity, which spreads features across the hypersphere. Following S3W and STSW, we replace the Gaussian kernel uniformity loss with FS3W, resulting in the objective

where zA, zB ∈ Rn×(d+1) are embeddings of two augmented views of the same input, projected onto the unit hypersphere Sd . The factor λ > 0 balances alignment and uniformity, and ν = U (Sd ) denotes the uniform hyperspherical prior.

Setup. Similar to [Tran et al., 2025], we train a ResNet18 on CIFAR-10 for 200 epochs with batch size 512, using SGD with lr = 0.05, momentum 0.9, and weight decay 10-3 . Positive pairs are generated by standard augmentations, including resizing/cropping, flipping, color jittering, and grayscale conversion [Zhang et al., 2025]. We then train a linear classifier on the learned features for 100 epochs using Adam with lr = 10-3 and decay 0.2 at epochs 60 and 80, following SSW. For STSW, we set L = 200 trees and k = 20 lines; other sliced distances use L = 200 projections. Following [Tran et al., 2024], RI-S3W and ARI-S3W use NR = 5 rotations from a pool of 100 random rotations.

Results. Table 4 shows that ARI-LAFS3W achieves the best linear evaluation accuracy on encoded features, while NFS3W reaches the highest accuracy on projected features on S9 . Compared with the original S3W variants, FS3W performs better on both encoded and projected features on CIFAR-10, demonstrating clear gains across settings. The visualizations in Fig. 3 shows that FS3W produces uniform and well-clustered embeddings on S2 , highlighting its strength in spherical representation learning.

6 Conclusion

This work introduces FS3W, a new approach for comparing probability distributions on the hypersphere. FS3W adapts a neural slicing fusion mechanism to refine directions adaptively using data-dependent information. By aligning directions with the discriminative geometric structures in the data distributions, FS3W provides a more expressive and reliable measure of distributional discrepancy, demonstrating strong performance in tasks like gradient flow, Earth density estimation, and self-supervised learning. Future work will focus on reducing the additional computational cost introduced by neural training and improving the efficiency of the method. The fusion mechanism is both general and modular, and it can be extended to the sliced Wasserstein distance in Euclidean spaces and other non-Euclidean settings, including hyperbolic spaces and manifolds. These extensions promise broader applications of data-adaptive sliced optimal transport and offer potential for further advancements in the field.

Acknowledgements

This work was supported by the NSFC under Grant Nos. 62276135, 62176124, U24A20330 and 62361166670, as well as by the Basic Research Program of Jiangsu under Grant No. BK20253028.