Stability of Wasserstein spaces under various convergence types.
problem Stability and finiteness of Wasserstein spaces over singular and non-singular spaces.
method Gromov--Hausdorff convergence and equivariant Gromov--Hausdorff convergence.
result Analogue of Perelman's stability theorem on Wasserstein spaces.
New analysis improves convergence guarantees for diffusion-based samplers in Wasserstein distance.
problem Improving convergence guarantees for diffusion-based generative models.
method Simple framework to analyze discretization, initialization, and score estimation errors.
result First Wasserstein convergence bound for the Heun sampler and improved results for Euler sampler.
Study shows convergence of stochastic gradient method for unregularized Wasserstein optimization.
problem Wasserstein distributionally robust optimization under potential distribution shifts.
method Regularized approximation with stochastic gradient methods, convergence analysis.
result Stochastic gradient method converges to subgradients of unregularized objective as regularization vanishes.
Study convergence and approximations of entropic regularized Wasserstein distances for Gaussian and RKHS measures.
problem Convergence and approximations of entropic regularized Wasserstein distances in Gaussian and RKHS settings.
method Analysis of convergence and finite sample approximations of entropic regularized Wasserstein distances in Gaussian and RKHS settings.
result Strictly weaker convergence in 2-Sinkhorn divergence for Gaussian measures compared to exact 2-Wasserstein distance.
Unified framework for analyzing convergence of RSAs using Wasserstein divergence.
problem Analyzing convergence of constant stepsize recursive stochastic algorithms (RSAs).
method Lifting RSA into a higher-dimensional space as a Markov chain and studying the distribution's contraction property with respect to Wasserstein divergence.
result RSAs' iterates' distribution converges to an invariant distribution under certain contraction properties.
Improved KL bounds and Wasserstein guarantees for diffusion flow matching under minimal conditions.
problem Theoretical convergence properties of Brownian motion based diffusion flow matching.
method Refined analysis under Kullback-Leibler and 2-Wasserstein distances.
result State-of-the-art scaling in KL convergence bounds under minimal conditions.
Paper analyzes convergence of proximal algorithm in metric spaces without geodesic convexity.
problem Analyzing convergence of proximal algorithm in general metric spaces.
method Analysis of the Wasserstein proximal algorithm without geodesic convexity assumption.
result Establishes unbiased and linear convergence rate for proximal algorithm under natural Wasserstein inequality.
Study on deep matrix factorization with Bures-Wasserstein loss, focusing on critical points and convergence.
problem Analyzing critical points and convergence of generative deep linear networks trained with Bures-Wasserstein loss.
method Characterization of critical points and minimizers of Bures-Wasserstein distance, analysis of Hessian at low-rank matrices, convergence results for gradient flow and descent.
result Established convergence results for gradient flow and finite step size gradient descent under certain assumptions.
New algorithm for computing Wasserstein barycenters with guarantees.
problem Computing Wasserstein barycenters with varying regularization strengths.
method Damped Sinkhorn iterations followed by exact maximization/minimization steps.
result First non-asymptotic convergence guarantees for approximating Wasserstein barycenters.
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.
A new robust metric compares distributions more accurately than existing methods.
problem Sensitivity to outliers and sampling discrepancy in Wasserstein distances.
method Introducing k-RPW, a partial p-Wasserstein distance.
result k-RPW converges faster to true distance and is more robust to outliers.
This paper studies convergence behavior of latent mixing measures that arise in finite and infinite mixture models, using transportation distances (i.e., Wasserstein metrics). The relationship between Wasserstein distances on the space of mixing measures and f-divergence functionals such as Hellinger and Kullback-Leibl…
Study shows rate of convergence for particle approximation of PDEs in Wasserstein space.
problem Analyzing convergence rates for particle approximations of PDEs in Wasserstein space.
method Backward stochastic differential equations techniques.
result Proved a rate of convergence of order 1/N for pathwise error and 1/sqrt(N) for L2-error on the derivative.
A new dynamical formulation of log-PCA captures local principal modes of geodesic variations.
problem Learning principal variations of random probability measures under Wasserstein geometry.
method Introducing a new dynamical formulation of log-PCA as a variational approach.
result Deriving a general statistical convergence rate for empirical WT-PCA.
Paper proves convergence of Gini index to equilibrium in Wasserstein distance.
problem Proving convergence of Gini index to equilibrium in Wasserstein distance.
method Analyzes Gini index as Lyapunov functional and proves convergence in Wasserstein distance.
result Proves convergence of Gini index to equilibrium in Wasserstein distance.
A new ParVI framework improves particle-based variational inference methods.
problem Non-trivial kernel design in particle-based variational inference methods.
method Proposes a generalized Wasserstein gradient descent (GWG) framework with broader regularizers.
result Demonstrates strong convergence guarantees and effectiveness on simulated and real data.
Develops a two-sample test using projected Wasserstein distance to handle high-dimensional data.
problem Testing whether two high-dimensional samples come from the same distribution.
method Optimal projection to find a low-dimensional linear mapping that maximizes the Wasserstein distance between projected probability distributions.
result Characterizes the convergence rate of the projected Wasserstein distance and presents practical algorithms.
PWGF escapes saddle points in nonconvex optimization.
problem Escaping saddle points in nonconvex optimization.
method PWGF uses noisy perturbations via Gaussian process to escape saddle points.
result PWGF achieves second-order optimality for nonconvex objectives.
This work studies the smooth 1-Wasserstein distance and its limit distribution in high dimensions.
problem Addressing the curse of dimensionality in empirical approximation.
method Conducts a statistical study including limit distribution, bootstrap consistency, and concentration inequalities.
result Derives a nondegenerate limit distribution for empirical SWD, contrasting with classic W 1 W_1 W 1 . Robust Q Q Q -learning for mean-field control under Wasserstein uncertainty
problem Mean-field control under Wasserstein uncertainty
method Quantization-and-projection scheme with Wasserstein dual reformulation
result Convergence and finite-time iteration bounds
Proof of convergence for multi-objective optimization using inverse reinforcement learning.
problem Proving convergence in multi-objective optimization problems.
method Wasserstein inverse reinforcement learning with projective subgradient method and gradient descent.
result Convergence of inverse reinforcement learning for multi-objective optimization.
WSFN overcomes saddle points for non-convex functionals in Wasserstein space.
problem Minimizing non-convex functionals over the Wasserstein space with saddle point avoidance.
method WSFN is a second-order method that preconditions the Wasserstein gradient to avoid saddle points.
result WSFN escapes saddle regions and reaches a global minimizer in polynomial time.
Paper analyzes convergence of ODE samplers in Wasserstein distances.
problem Limited theoretical understanding of convergence properties of probability flow ODEs.
method Convergence analysis for general probability flow ODEs in 2-Wasserstein distance.
result First non-asymptotic convergence analysis for probability flow ODE samplers.
Improved convergence rates for MLE in mixture models using penalized log-likelihood.
problem Convergence rates for MLE in finite mixture models.
method Penalizing log-likelihood to discourage vanishing mixing weights, using Wasserstein distance and new loss functions.
result Improved convergence rates for some mixture components, faster than traditional methods.
The paper explores multidimensional critic output in GANs, improving convergence and diversity.
problem Underexplored in GANs literature, multidimensional critic output.
method Generalized Wasserstein GAN framework, SRVT block, maximal p-centrality discrepancy.
result High-dimensional critic output improves GAN performance in convergence and diversity.
This work improves convergence guarantees for unadjusted HMC in KL and Rényi divergences.
problem Understanding convergence properties of unadjusted HMC in divergences like KL and Rényi.
method One-shot couplings to establish regularization and lift convergence bounds.
result Quantitative control of relative density mismatch and warm-start requirements.
The Sinkhorn flow converges to a Wasserstein mirror gradient flow from the Sinkhorn algorithm.
problem Optimizing joint distributions using the Sinkhorn algorithm.
method Wasserstein mirror gradient flow derived from the Sinkhorn algorithm.
result The Sinkhorn flow converges to a Wasserstein mirror gradient flow.
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.
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.
The paper introduces a new ODE approach to improve Wasserstein GANs.
problem Improving Wasserstein GANs for better training results.
method Derives an ODE representing the gradient flow of Wasserstein-1 loss and proposes a new model W1-FE.
result W1-FE outperforms WGAN in training experiments across various dimensions.
The paper extends graph-based semi-supervised learning to infinite-dimensional Wasserstein space.
problem Graph-based semi-supervised learning in high-dimensional data.
method Laplace Learning in the Wasserstein space, proving variational convergence and characterizing the Laplace-Beltrami operator.
result Consistent classification performance in high-dimensional settings.
Aggregates probability models using Wasserstein space and variational approach.
problem Model aggregation in the Wasserstein space of distributions.
method Data-driven calibration framework based on Γ Γ Γ -convergence. result Empirical minimizers converge to the minimizers of the actual problem.
New method speeds up optimization over probability measures.
problem High computational overhead in optimizing probability measures.
method Randomized coordinate descent on Wasserstein space.
result Significant speedups over full-gradient methods.
Minimum expected distance estimation (MEDE) algorithms have been widely used for probabilistic models with intractable likelihood functions and they have become increasingly popular due to their use in implicit generative modeling (e.g. Wasserstein generative adversarial networks, Wasserstein autoencoders). Emerging fr…
DDPMs are robust to noisy score estimates and achieve optimal convergence rates in Wasserstein-2 distance.
problem Evaluating the quality of DDPMs in Wasserstein distance with noisy score estimates.
method Established finite-sample guarantees in Wasserstein-2 distance for DDPMs, considering noisy score estimates.
result Optimal convergence rates in Wasserstein-2 distance for DDPMs, matching Gaussian case.
Develops calculus on Wasserstein spaces for Riemannian manifolds.
problem Characterizing and understanding the geometry of Wasserstein spaces.
method Intrinsic formalism for topology, smooth structure, and Riemannian geometry of Wasserstein spaces.
result Wasserstein spaces of closed manifolds are geodesically convex.
WGANs use optimal 1-Wasserstein distance to generate distributions.
problem Characterize geometrical properties of generated distributions.
method Analyze WGANs in finite and asymptotic regimes, focusing on univariate latent space.
result WGANs can approach target distribution with optimal 1-Wasserstein distance as sample size increases.
pHMC converges on infinite-dimensional spaces with bounds.
problem Convergence of pHMC on Hilbert spaces.
method Coupling of two pHMC copies, adapted from arXiv:1805.00452.
result Proven convergence bounds in 1-Wasserstein distance.
Improved convergence rates for Stein Variational Gradient Descent in finite-particle settings.
problem Improving convergence rates for Stein Variational Gradient Descent in finite-particle settings.
method Analyzing the time derivative of relative entropy and splitting it into dominant and smaller parts.
result Finite-particle convergence rates of order 1/\sqrt{N} for Kernelized Stein Discrepancy and Wasserstein-2 metrics.
The paper establishes convergence guarantees for SGMs in 2-Wasserstein distance.
problem Establishing convergence guarantees for SGMs in 2-Wasserstein distance.
method Assuming accurate score estimates and smooth log-concave data distribution, the paper specializes its result to several concrete SGMs with specific forward processes modeled by stochastic differential equations.
result Obtained an upper bound on the iteration complexity for each model and a lower bound for Gaussian data distribution.
Wasserstein-GANs have been introduced to address the deficiencies of generative adversarial networks (GANs) regarding the problems of vanishing gradients and mode collapse during the training, leading to improved convergence behaviour and improved image quality. However, Wasserstein-GANs require the discriminator to be…
This research analyzes and accelerates score-based diffusion models using discretization and Hessian information.
problem Theoretical foundations and convergence analysis of score-based diffusion models.
method Investigation of various discretization schemes, including Euler, exponential integrators, and midpoint randomization. Proposal of an accelerated sampler based on local linearization method.
result Hessian-based approach achieves faster convergence rates of order $\widetilde{\mathcal{O}}\left(\frac{1}{\varepsilon}
ight)$ , significantly improving upon vanilla diffusion models.
This work improves SGMs' convergence guarantees for semiconvex distributions with discontinuous gradients.
problem Establishing convergence guarantees for SGMs under weak regularity conditions.
method Developed non-asymptotic Wasserstein-2 convergence analysis for SGMs targeting semiconvex distributions with discontinuous gradients.
result Achieved optimal dependence of O ( d ) O(\sqrt{d}) O ( d ) on data dimension d d d and convergence rate of order one. Optimizes sample reweighting to match laws under covariate shift using Wasserstein distance.
problem Matching laws of samples with different distributions under covariate shift.
method Minimizes Wasserstein distance between empirical measures of samples using Nearest Neighbors weights.
result Consistent reweighting leads to asymptotic convergence of empirical measures.
The study provides statistical theory for WGANs in time series forecasting.
problem Statistical analysis of WGANs for time series forecasting.
method Statistical theory and upper bounds for excess Bayes risk, weak convergence, and confidence intervals.
result Developed confidence intervals for time series forecasting using WGANs.
Unified theory of optimal transport for random measures.
problem Statistical uncertainty in optimal transport.
method Constructing L 2 L^2 L 2 over Wasserstein space for random probability measures. result Unified treatment of random optimal transport and principled inference.
New KSDs control moments in approximations, improving diagnostics and tests.
problem Inability of standard KSDs to control moment convergence.
method Developed alternative diffusion KSDs under sufficient conditions.
result First KSDs to exactly characterize q-Wasserstein convergence.
A new method for Bayesian inference tackles high-dimensional problems.
problem Bayesian inference in high-dimensional settings with kernel density estimation issues.
method Projected Wasserstein gradient descent (pWGD) method to overcome curse of dimensionality.
result pWGD method effectively addresses high-dimensional Bayesian inference problems.