We propose fast approximations for the generalized sliced-Wasserstein distance.
problem Efficient approximation of the generalized sliced-Wasserstein distance in high dimensions.
method Deterministic approximations using random projections and concentration of measure results.
result One-dimensional projections of high-dimensional random vectors are approximately Gaussian.
A new slicing method speeds up sliced Wasserstein estimation.
problem Efficiently estimating sliced Wasserstein distance.
method Random-Path Projecting Direction (RPD) for fast sampling.
result RPSW and IWRPSW show favorable performance in training generative models.
New PAC-Bayesian bounds improve Sliced-Wasserstein distances.
problem Improving statistical properties of Sliced-Wasserstein distances.
method Leveraging PAC-Bayesian theory to provide bounds and learning procedures.
result PAC-Bayesian generalization bounds for adaptive SW distances.
A new method optimizes projection directions for sliced Wasserstein distances.
problem Finding informative projecting directions for sliced Wasserstein distances is computationally expensive.
method Amortized projection optimization to predict directions efficiently.
result Proposed amortized models improve generative modeling performance.
A new method for comparing image probability measures using convolution operators.
problem Efficiently comparing images using conventional sliced Wasserstein methods.
method Proposed convolution sliced Wasserstein (CSW) methods with stride, dilation, and non-linear activation.
result CSW demonstrates favorable performance over conventional sliced Wasserstein in image comparison and deep generative modeling.
Extends SW and GSW to compare heterogeneous joint distributions.
problem Limited applicability of SW and GSW to heterogeneous joint distributions.
method Introduces HHRT and PGRT to extend SW and GSW.
result H2SW distance for heterogeneous joint distributions.
Proposes a new distance metric for multi-marginal optimal transport.
problem Computational scalability in multi-marginal optimal transport.
method Random one-dimensional projections to construct sliced multi-marginal Wasserstein distance.
result Sliced multi-marginal Wasserstein distance is a metric with dimension-free sample complexity.
Upper bound for max-sliced 2-Wasserstein distance between measures.
problem Estimating distance between probability measures and their empirical counterparts.
method Same technique as previous work, upper bound approach.
result Upper bound for expected max-sliced 2-Wasserstein distance.
A new distance measure balances projection exploration and informativeness.
problem Inefficient and incomplete projection sampling in existing sliced-Wasserstein distances.
method Proposes Distributional Sliced-Wasserstein (DSW) that optimally balances projection exploration and informativeness.
result DSW generalizes Max-SW and can be computed efficiently.
A new approach simplifies Sliced-Wasserstein distances to improve learning performance.
problem The concentration of measure phenomenon makes random projections uninformative in high dimensions.
method Propose rescaling the 1D Wasserstein distance to make all slices equally informative.
result The classical Sliced-Wasserstein, properly configured, can match or surpass complex variants.
New Sliced-Wasserstein distances for non-Euclidean data.
problem Computational burden of Wasserstein distance on non-Euclidean manifolds.
method Derive Sliced-Wasserstein distances and flows on Cartan-Hadamard manifolds.
result General constructions and non-parametric schemes for minimizing new distances.
A new sliced IGW distance for Gromov-Wasserstein alignment.
problem Scalability issues in Gromov-Wasserstein alignment for high-dimensional problems.
method Proposed a sliced IGW distance with rotational invariance.
result Natural rotational invariance of the sliced IGW distance.
This paper introduces repulsive Monte Carlo methods for computing the sliced Wasserstein distance.
problem Computing the integral of a function on the unit sphere using Monte Carlo methods.
method The approach involves using determinantal point processes and repelled point processes to create quadratures for the sliced Wasserstein distance.
result The UnifOrtho estimator is recommended for the computation of the sliced Wasserstein distance in large dimensions.
Researchers developed a differentially private method for computing Wasserstein distances.
problem Computing divergences between distributions while preserving privacy.
method They focused on the Sliced Wasserstein Distance and added Gaussian perturbations to make it differentially private.
result They introduced a new differentially private distance, the Smoothed Sliced Wasserstein Distance, which performs well in generative models and domain adaptation.
Proposes MFSWB for marginal fairness in SWB, improving efficiency and performance.
problem Achieving marginal fairness in SWB averaging.
method Defining MFSWB as a constrained SWB problem, proposing two surrogate problems and a new slicing distribution.
result Surrogate MFSWB problems effectively minimize distances to marginals and encourage marginal fairness.
In this paper we study generative modeling via autoencoders while using the elegant geometric properties of the optimal transport (OT) problem and the Wasserstein distances. We introduce Sliced-Wasserstein Autoencoders (SWAE), which are generative models that enable one to shape the distribution of the latent space int…
New autoencoder improves latent space learning by optimizing sliced Gromov-Wasserstein discrepancies.
problem Improving inner discrepancy between prior and posterior distributions in autoencoders.
method Proposed spherical sliced fused Gromov Wasserstein (SSFG) and variants (MSSFG, PSSFG) to find important directions.
result New autoencoders achieve favorable performance in latent manifold learning, image generation, and reconstruction.
A new spherical Sliced-Wasserstein distance for data on spheres.
problem Defining Wasserstein distance on manifolds, especially spheres.
method Closed-form solutions of the Wasserstein distance on the circle and a new spherical Radon transform.
result A novel spherical Sliced-Wasserstein (SW) discrepancy for data on spheres.
Proposes an energy-based sliced Wasserstein distance for improved probability measure comparison.
problem Inefficiencies and limitations in existing sliced Wasserstein distance approaches.
method Introduces an energy-based slicing distribution for better performance and stability.
result Demonstrates superior performance of the EBSW distance in various applications.
A new metric HSW derived from hierarchical Radon Transform addresses computational bottlenecks in sliced Wasserstein.
problem Computational inefficiency of sliced Wasserstein in high-dimensional settings with few supports.
method Hierarchical Radon Transform (HRT) and bottleneck projections to reduce projection number.
result HSW metric derived from HRT is computationally efficient and maintains metric properties.
New slicing methods speed up Gaussian mixture Wasserstein distance computations.
problem High computational cost of the mixture Wasserstein distance.
method Slicing-based approximations to reduce computational complexity.
result Significant reduction in computational complexity while preserving key properties.
Wasserstein distances are increasingly used in a wide variety of applications in machine learning. Sliced Wasserstein distances form an important subclass which may be estimated efficiently through one-dimensional sorting operations. In this paper, we propose a new variant of sliced Wasserstein distance, study the use …
Generative model improved using Liouville PDE-based sliced-Wasserstein flow.
problem Improving generative models for fair regression.
method Transformed sliced-Wasserstein flow into Liouville PDE-based formalism, handling density estimation with normalizing flows of neural ODE.
result Outperforms in convergence and fairness with reduced variance.
A new method for fast optimal transport using sliced Wasserstein generalized geodesics.
problem Computing optimal transport distances efficiently and accurately.
method Proposes a new proxy of squared Wasserstein distance based on one-dimensional projections.
result min-SWGG is an upper bound of Wasserstein distance with similar computational complexity.
This paper improves normalizing flows by combining MLE and sliced-Wasserstein distance for better data fidelity.
problem Normalizing flows struggle with generating realistic data and detecting out-of-distribution data.
method Proposes a hybrid objective function combining MLE and sliced-Wasserstein distance.
result Shows better generative abilities and lower likelihood of out-of-distribution data.
Generative adversarial nets (GANs) and variational auto-encoders have significantly improved our distribution modeling capabilities, showing promise for dataset augmentation, image-to-image translation and feature learning. However, to model high-dimensional distributions, sequential training and stacked architectures …
A new Wasserstein distance method for comparing incomparable distributions.
problem Comparing distributions that are not supported on the same metric space.
method Distributional slicing, embeddings, and closed-form computation of Wasserstein distance.
result HWD preserves properties like rotation-invariance and can be efficiently learned.
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 vario…
Sharp bounds for max-sliced Wasserstein distances derived for empirical distributions.
problem Estimating the expected max-sliced Wasserstein distance between a probability measure and its empirical distribution.
method Banach space version and operator norm approach for upper bounds.
result Upper bounds for max-sliced Wasserstein distances are essentially matching and sharp up to a log factor.
New hyperbolic sliced-Wasserstein distances derived for efficient comparison.
problem Efficient comparison of distributions in hyperbolic spaces.
method Projections on geodesics or horospheres to derive novel sliced-Wasserstein distances.
result Novel hyperbolic sliced-Wasserstein distances are more computationally efficient.
This work improves scalability of Wasserstein distances in high dimensions.
problem Scalability issues in computing Wasserstein distances in high dimensions.
method Empirical convergence rates, robustness to data contamination, and computational methods.
result Established fast rates and robust estimation risks for sliced Wasserstein distances.
Paper optimizes WGAN parameters for non-Gaussian data.
problem Optimizing parameters for non-Gaussian data in WGAN.
method Characterization of optimal solutions for population WGAN beyond LQG setting, using sliced Wasserstein framework.
result Closed-form optimal parameters for non-linear activation functions and non-Gaussian data derived.
New method for summarizing Bayesian mixture models using sliced Wasserstein distances.
problem Estimating the mixing measure in nonparametric Bayesian mixture models.
method Decision-theoretic approach using sliced Wasserstein distances for Gaussian mixtures.
result Effective estimation of the mixing measure and mixture density.
Motivated by the growing popularity of variants of the Wasserstein distance in statistics and machine learning, we study statistical inference for the Sliced Wasserstein distance--an easily computable variant of the Wasserstein distance. Specifically, we construct confidence intervals for the Sliced Wasserstein distanc…
A method for fast estimation of Wasserstein distances using sliced Wasserstein distances.
problem Efficiently computing Wasserstein distances for multiple pairs of distributions.
method Regression on sliced Wasserstein distances to predict true Wasserstein distances.
result The proposed method provides a better approximation of Wasserstein distance than state-of-the-art models, especially in low-data regimes.
Optimal transport (\OT) theory defines a powerful set of tools to compare probability distributions. \OT~suffers however from a few drawbacks, computational and statistical, which have encouraged the proposal of several regularized variants of OT in the recent literature, one of the most notable being the \textit{slice…
Introduces MSW distances to improve SW metrics.
problem Redundant projections in SW distance.
method Imposes Markov structure on projecting directions.
result MSW distances improve SW metrics.
A new variational inference method using sliced Wasserstein distance is proposed.
problem The inefficiency and unreasonable properties of Kullback-Leibler divergence.
method Minimizing sliced Wasserstein distance, a valid metric from optimal transport.
result The proposed method approximates the unnormalized distribution efficiently and without requiring a tractable density function.
Improves point-cloud reconstruction by optimizing projections with self-attention.
problem Inefficient and non-metric projection methods for sliced Wasserstein distances.
method Proposes distributional sliced Wasserstein distance with self-attention for permutation-invariant and metric optimization.
result Self-attention amortized distributional projection optimization achieves better performance in point-cloud reconstruction.
Proposes QMC-based QSW for 3D SW distance.
problem Intractable SW distance in 3D.
method Quasi-Monte Carlo (QMC) for QSW approximations.
result QMC-based QSW improves SW estimation.
Researchers find Busemann functions in Wasserstein space, enabling efficient projections and distances.
problem Defining Busemann functions in Wasserstein space for efficient data projections and distances.
method Investigated existence and computation of Busemann functions in Wasserstein space, establishing closed-form expressions for specific cases.
result Explicit projection schemes for probability distributions on \(\mathbb{R}\) enable novel Sliced-Wasserstein distances over Gaussian mixtures and labeled datasets.
Paper proves convergence of measure transfer schemes using slicing and matching.
problem Iterative schemes for measure transfer and approximation problems.
method Slicing-and-matching procedure, stochastic gradient descent on Wasserstein space.
result Almost sure convergence proof for stochastic slicing-and-matching schemes.
This paper proposes a new method to solve functional minimization problems in probability distributions using sliced-Wasserstein gradient flows.
problem Solving functional minimization problems in high-dimensional probability distributions is computationally challenging.
method The paper introduces a new approach using sliced-Wasserstein gradient flows to approximate the Jordan-Kinderlehrer-Otto (JKO) scheme, parameterizing densities with generative models.
result The proposed method is more flexible and computationally tractable compared to existing methods like JKO-ICNN.
Approximate Bayesian Computation (ABC) is a popular method for approximate inference in generative models with intractable but easy-to-sample likelihood. It constructs an approximate posterior distribution by finding parameters for which the simulated data are close to the observations in terms of summary statistics. T…
This work investigates the properties of Gaussian-smoothed sliced divergences for comparing distributions.
problem Comparing probability distributions while preserving privacy.
method Investigates the theoretical properties of Gaussian-smoothed sliced Wasserstein distance and generalized versions.
result Gaussian smoothed sliced Wasserstein distance converges with a rate of \(O(n^{-1/2})\).
Paper introduces a differentially private generative model using gradient flow and sliced Wasserstein distance.
problem Protecting privacy in sensitive training data for generative models.
method Gradient flow in the space of probability measures, Gaussian-smoothed Sliced Wasserstein Distance, and numerical scheme for SDE.
result Demonstrates higher-fidelity data generation at low privacy budget compared to existing methods.
A new method approximates the Sliced-Wasserstein distance without random projections.
problem Efficiently approximating the Sliced-Wasserstein distance for machine learning applications.
method Utilizing the concentration of measure phenomenon to develop a deterministic approximation.
result The approximation error goes to zero as the dimension increases, under a weak dependence condition.
The paper studies properties of Sliced Wasserstein energy for discrete measures.
problem Optimizing discrete probability measures using Sliced Wasserstein loss.
method Investigates the regularity and optimisation properties of the Sliced Wasserstein energy and its Monte-Carlo approximation.
result Convergence results on the critical points of Monte-Carlo approximations to the Sliced Wasserstein energy.