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.

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.

However, finding these directions usually requires an iterative optimization procedure over the space of projecting directions, which is computationally expensive. Moreover, the computational issue is even more severe in deep learning applications, where computing the distance between two mini-batch probability measures is repeated several times.

However, this approach has two limitations.

Sliced optimal transport (SOT) or sliced Wasserstein (SW) distance is widely recognized for its statistical and computational scalability. In this work, we further enhance the computational scalability by proposing the first method for computing SW from sample streams, called \emph{streaming sliced Wasserstein} (Stream-SW). To define Stream-SW, we first introduce the streaming computation of the one-dimensional Wasserstein distance. Since the one-dimensional Wasserstein (1DW) distance has a closed-form expression, given by the absolute difference between the quantile functions of the compared distributions, we leverage quantile approximation techniques for sample streams to define the streaming 1DW distance. By applying streaming 1DW to all projections, we obtain Stream-SW. The key advantage of Stream-SW is its low memory complexity while providing theoretical guarantees on the approximation error. We demonstrate that Stream-SW achieves a more accurate approximation of SW than random subsampling, with lower memory consumption, in comparing Gaussian distributions and mixtures of Gaussians from streaming samples. Additionally, we conduct experiments on point cloud classification, point cloud gradient flows, and streaming change point detection to further highlight the favorable performance of Stream-SW.

Determining which parameters of a non-linear model could best describe a set of experimental data is a fundamental problem in science and it has gained much traction lately with the rise of complex large-scale simulators (a.k.a. black-box simulators). The likelihood of such models is typically intractable, which is why classical MCMC methods can not be used. Simulation-based inference (SBI) stands out in this context by only requiring a dataset of simulations to train deep generative models capable of approximating the posterior distribution that relates input parameters to a given observation. In this work, we consider a tall data extension in which multiple observations are available and one wishes to leverage their shared information to better infer the parameters of the model. The method we propose is built upon recent developments from the flourishing score-based diffusion literature and allows us to estimate the tall data posterior distribution simply using information from the score network trained on individual observations. We compare our method to recently proposed competing approaches on various numerical experiments and demonstrate its superiority in terms of numerical stability and computational cost.