Inthis paper,wefirst clarify themathematical connection between the SW distance and the Radon transform. We then utilize the generalized Radon transform to define a new family of distances for probability measures, which we call generalized sliced-Wasserstein (GSW) distances.

Sliced Wasserstein (SW) and Generalized Sliced Wasserstein (GSW) have been widely used in applications due to their computational and statistical scalability. However, the SW and the GSW are only defined between distributions supported on a homogeneous domain. This limitation prevents their usage in applications with heterogeneous joint distributions with marginal distributions supported on multiple different domains. Using SW and GSW directly on the joint domains cannot make a meaningful comparison since their homogeneous slicing operator, i.e., Radon Transform (RT) and Generalized Radon Transform (GRT) are not expressive enough to capture the structure of the joint supports set. To address the issue, we propose two new slicing operators, i.e., Partial Generalized Radon Transform (PGRT) and Hierarchical Hybrid Radon Transform (HHRT). In greater detail, PGRT is the generalization of Partial Radon Transform (PRT), which transforms a subset of function arguments non-linearly while HHRT is the composition of PRT and multiple domain-specific PGRT on marginal domain arguments. By using HHRT, we extend the SW into Hierarchical Hybrid Sliced Wasserstein (H2SW) distance which is designed specifically for comparing heterogeneous joint distributions. We then discuss the topological, statistical, and computational properties of H2SW. Finally, we demonstrate the favorable performance of H2SW in 3D mesh deformation, deep 3D mesh autoencoders, and datasets comparison.

The Wasserstein distance and its variations, e.g., the sliced-Wasserstein (SW) distance, have recently drawn attention from the machine learning community. The SW distance, specifically, was shown to have similar properties to the Wasserstein distance, while being much simpler to compute, and is therefore used in various applications including generative modeling and general supervised/unsupervised learning. In this paper, we first clarify the mathematical connection between the SW distance and the Radon transform. We then utilize the generalized Radon transform to define a new family of distances for probability measures, which we call generalized sliced-Wasserstein (GSW) distances. We further show that, similar to the SW distance, the GSW distance can be extended to a maximum GSW (max-GSW) distance. We then provide the conditions under which GSW and max-GSW distances are indeed proper metrics. Finally, we compare the numerical performance of the proposed distances on the generative modeling task of SW flows and report favorable results.

Sliced Wasserstein (SW) and Generalized Sliced Wasserstein (GSW) have been widely used in applications due to their computational and statistical scalability.

Sliced Wasserstein (SW) and Generalized Sliced Wasserstein (GSW) have been widely used in applications due to their computational and statistical scalability.

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The results from one experiment are presented in Figure 2. We plot the resulting variational approximation of the posterior for The conclusions we draw from this experiment are the same as those in Section 4. With the optimal On the left-hand side of Figure 3, we also report the computation times of the methods, and we highlight that the true posterior takes much longer than any of the variational approximations. Again, similar conclusions can be drawn as in the previous sections. Computed T omography (CT) Imaging and Medical Single Photon Emission Computed T omography (SPECT): In CT scans, X-ray measurements are taken from different angles around a patient, and the Radon transform is used to reconstruct a cross-sectional image (slice) of the patient's body. This helps doctors visualize internal structures and diagnose various medical conditions. The Radon transform is used in the image reconstruction process for SPECT.

Tree-Sliced methods have recently emerged as an alternative to the traditional Sliced Wasserstein (SW) distance, replacing one-dimensional lines with tree-based metric spaces and incorporating a splitting mechanism for projecting measures. This approach enhances the ability to capture the topological structures of integration domains in Sliced Optimal Transport while maintaining low computational costs. Building on this foundation, we propose a novel nonlinear projectional framework for the Tree-Sliced Wasserstein (TSW) distance, substituting the linear projections in earlier versions with general projections, while ensuring the injectivity of the associated Radon Transform and preserving the well-definedness of the resulting metric. By designing appropriate projections, we construct efficient metrics for measures on both Euclidean spaces and spheres. Finally, we validate our proposed metric through extensive numerical experiments for Euclidean and spherical datasets. Applications include gradient flows, self-supervised learning, and generative models, where our methods demonstrate significant improvements over recent SW and TSW variants.

Diabetic retinopathy is a serious ocular complication that poses a significant threat to patients' vision and overall health. Early detection and accurate grading are essential to prevent vision loss. Current automatic grading methods rely heavily on deep learning applied to retinal fundus images, but the complex, irregular patterns of lesions in these images, which vary in shape and distribution, make it difficult to capture subtle changes. This study introduces RadFuse, a multi-representation deep learning framework that integrates non-linear RadEx-transformed sinogram images with traditional fundus images to enhance diabetic retinopathy detection and grading. Our RadEx transformation, an optimized non-linear extension of the Radon transform, generates sinogram representations to capture complex retinal lesion patterns. By leveraging both spatial and transformed domain information, RadFuse enriches the feature set available to deep learning models, improving the differentiation of severity levels. We conducted extensive experiments on two benchmark datasets, APTOS-2019 and DDR, using three convolutional neural networks (CNNs): ResNeXt-50, MobileNetV2, and VGG19. RadFuse showed significant improvements over fundus-image-only models across all three CNN architectures and outperformed state-of-the-art methods on both datasets. For severity grading across five stages, RadFuse achieved a quadratic weighted kappa of 93.24%, an accuracy of 87.07%, and an F1-score of 87.17%. In binary classification between healthy and diabetic retinopathy cases, the method reached an accuracy of 99.09%, precision of 98.58%, and recall of 99.6%, surpassing previously established models. These results demonstrate RadFuse's capacity to capture complex non-linear features, advancing diabetic retinopathy classification and promoting the integration of advanced mathematical transforms in medical image analysis.