This paper bridges variational inference and Wasserstein gradient flows.
problem Combining variational inference and Wasserstein gradient flows for more efficient approximations.
method Recasting Bures-Wasserstein gradient flow as a Euclidean gradient flow and using path-derivative gradient estimator.
result A new gradient estimator for f-divergences that can be implemented using machine learning libraries. Optimal neural network approximation for Wasserstein gradient direction via convex optimization.
problem Approximating Wasserstein gradient direction with limited data.
method Two-layer networks with squared-ReLU activations, SDP relaxation.
result Optimal approximation of Wasserstein gradient direction in two-layer networks.
Wasserstein gradient boosting predicts probability distributions for supervised learning.
problem Distribution-valued supervised learning where outputs are probability distributions.
method Fits a new weak learner to Wasserstein gradients of loss functionals of probability distributions.
result Superior performance in probabilistic prediction compared to existing methods.
Forward-Euler fails for simulating Wasserstein gradient flows with KL divergence.
problem Simulating Wasserstein gradient flows with forward-Euler discretization fails for KL divergence.
method Forward-Euler discretization for Wasserstein gradient flows with KL divergence.
result Forward-Euler discretization can be incorrect for Wasserstein gradient flows with KL divergence.
This work explores gradient flows and Riemannian structure in Gromov-Wasserstein geometry for data with global structure.
problem Suitable geometry for tasks requiring preservation of global data structure.
method Study of gradient flows and Riemannian structure in Gromov-Wasserstein geometry for distributions on \(\mathbb{R}^d\).
result Established a Benamou-Brenier-like formula for IGW and derived the IGW gradient.
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.
This work proposes a new method for variational inference using Wasserstein gradient descent.
problem Optimizing variational parameters to match a true posterior distribution.
method Reinterpreting VI as an optimization problem over a variational parameter space, using Wasserstein gradient descent.
result The proposed Wasserstein gradient descent can be seen as a generalization of existing optimization techniques in VI.
We present a framework for Nesterov's accelerated gradient flows in probability space to design efficient mean-field Markov chain Monte Carlo (MCMC) algorithms for Bayesian inverse problems. Here four examples of information metrics are considered, including Fisher-Rao metric, Wasserstein-2 metric, Kalman-Wasserstein m…
A new method for Gaussian filtering using gradient flows and Wasserstein metrics.
problem Approximating Gaussian and mixture-of-Gaussians filtering for complex systems.
method Variational approximation via gradient-flow representation on Wasserstein metric space.
result Competitive performance in posterior representation and parameter estimation for systems with multiplicative noise and multi-modal distributions.
Algorithm samples from Wasserstein barycenter of measures.
problem Sampling from Wasserstein barycenter of measures.
method Gradient flow of multimarginal formulation with penalization.
result Algorithm samples close to Wasserstein barycenter.
Paper analyzes inclusive KL inference using Wasserstein gradient flows.
problem Analyzing inclusive KL inference with mathematical tools.
method Gradient flows derived from PDE analysis.
result Unified view of existing sampling algorithms as inclusive-KL inference.
We present a novel approximate inference method for diffusion processes, based on the Wasserstein gradient flow formulation of the diffusion. In this formulation, the time-dependent density of the diffusion is derived as the limit of implicit Euler steps that follow the gradients of a particular free energy functional.…
The paper studies scaling limits of Wasserstein metrics on Gaussian mixture models.
problem Understanding the scaling limits of Wasserstein metrics on Gaussian mixture models.
method Scaling limit approach on Gaussian mixture models, including inhomogeneous and extended models.
result Existence of the limit of the Wasserstein metric after renormalization for GMMs with zero variance.
MWGraD solves multi-objective distributional optimization using particle-based gradient descent.
problem Simultaneously minimize multiple objective functionals over probability distributions.
method Iterative particle-based algorithm MWGraD, estimating and aggregating Wasserstein gradients.
result Demonstrates effectiveness on synthetic and real-world datasets.
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.
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.
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.
A new algorithm for minimizing functions on Wasserstein space.
problem Discretization of continuous Wasserstein gradient flows in machine learning.
method Forward-Backward discretization scheme for minimizing functions with smooth and nonsmooth components.
result The FB scheme converges similarly to proximal gradient algorithms in Euclidean spaces.
The paper analyzes rates for a modified gradient descent method using Stein variational gradients.
problem Improving the accuracy of gradient descent methods for complex target distributions.
method Derives finite-particle rates for regularized Stein variational gradient descent (R-SVGD).
result Establishes explicit non-asymptotic bounds for time-averaged empirical measures.
We study the Wasserstein natural gradient in parametric statistical models with continuous sample spaces. Our approach is to pull back the L2-Wasserstein metric tensor in the probability density space to a parameter space, equipping the latter with a positive definite metric tensor, under which it becomes a Riemanni…
Improved sampling method using regularized Stein Variational Gradient Flow.
problem Improving the accuracy of sampling methods in machine learning.
method Proposed Regularized Stein Variational Gradient Flow to interpolate between SVGD and Wasserstein Gradient Flow.
result Established theoretical properties and provided preliminary numerical evidence of improved performance.
Paper develops a generative model using Wasserstein-2 loss.
problem Creating realistic data samples from limited data.
method Uses a distribution-dependent ODE with a gradient flow for W2 loss.
result The method converges to the true data distribution exponentially.
New method uses diffusion models to solve inverse problems.
problem Solving ill-posed inverse problems with powerful priors.
method Formulate posterior sampling as a regularized Wasserstein gradient flow in latent space.
result Demonstrates improved performance on standard benchmarks.
Paper introduces geometry-aware normalizing flows for improved causal inference.
problem Disparity between sample and population distributions in causal inference.
method Integrates continuous normalizing flows with parametric submodels, employing Wasserstein gradient flows and optimal transport.
result Significantly reduces parameter estimation bias and variance in finite-sample settings.
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.
Gradient flows on distributions of distributions for machine learning tasks.
problem Designing gradient flows for datasets of probability distributions.
method Representing classes as conditional distributions, modeling datasets as mixture distributions, using Wasserstein over Wasserstein (WoW) distance and gradients.
result Demonstrated gradient flows for dataset transfer and distillation tasks.
We reduce variance in Bures-Wasserstein variational inference.
problem High variance in Monte Carlo approximations of Bures-Wasserstein gradients.
method Control variates to reduce variance in the forward step.
result Proposed estimator reduces variance by orders of magnitude.
New discretization scheme for Wasserstein gradient flows using Schrödinger bridges.
problem Computing Wasserstein gradient flows efficiently and without score functions.
method Iterated Schrödinger bridge approximation with particle-based Sinkhorn algorithm.
result The scheme converges to Wasserstein gradient flows for certain flows, including heat flow.
Improved VI with Price's gradient estimator for target log-density.
problem Approximating target distributions from unnormalized log-densities.
method Stochastic gradient-based variational inference with Price's gradient estimator.
result Identifies Price's gradient as the key to WVI's superior performance.
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.
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.
Paper presents a new method to learn unnormalized models efficiently.
problem Scalability issues in score matching for flexible unnormalized models.
method Connects learning objectives to Wasserstein gradient flows for scalability.
result Demonstrates improved learning of unnormalized models on manifolds.
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 for learning gradient flows from population dynamics.
problem Reconstructing population dynamics from limited data.
method Residual approach to enforce continuity equations, combining with data-fitting divergence.
result Demonstrated state-of-the-art performance across trajectory inference benchmarks.
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.
New method optimizes black-box functions using generative models and Wasserstein distance.
problem Optimizing black-box functions with stochastic responses in high dimensions.
method Deep generative surrogate models and Wasserstein distance for uncertainty estimation.
result Method outperforms state-of-the-art methods in robustness to function shape and stochasticity.
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 method for scalable barycenter computation using Wasserstein gradient flows.
problem Scalability and integration of label information in barycenter computation.
method Gradient flows in Wasserstein space, time discretization, mini-batch optimal transport, modular regularization, task-aware functions, supervised information integration.
result Empirically validated new state-of-the-art barycenter solver with labeled barycenters outperforming unlabeled ones.
DDEQs extend DEQs to discrete measure inputs using Wasserstein gradient flows.
problem Applying DEQs to discrete measure inputs like sets or point clouds.
method Wasserstein gradient flows for finding fixed points of discrete measures under permutation-invariance.
result DDEQs can compete with state-of-the-art models in tasks like point cloud classification and completion.
Wasserstein GANs with Gradient Penalty compute a different optimal transport problem called congested transport.
problem Training generative models to produce high-quality synthetic data.
method Wasserstein GANs with Gradient Penalty (WGAN-GP) approach to calculate the Wasserstein 1 distance.
result WGAN-GP computes the minimum of the congested transport problem, not the Wasserstein 1 distance.
Analyzed a generative model framework through Wasserstein Gradient Flow.
problem Generative modeling challenges.
method Wasserstein Gradient Flow (WGF) interpretation of Drifting Models (GMD).
result Different algorithms correspond to specific limiting points of WGFs on various divergences.
Error estimates found between SGD with momentum and Langevin diffusion.
problem Quantifying the difference between SGD with momentum and Langevin diffusion.
method Established error estimates using 1-Wasserstein and total variation distances.
result Quantitative error estimates between SGD with momentum and underdamped Langevin diffusion.
A new gradient flow framework for distributionally robust optimization.
problem Optimizing under uncertainty with worst-case distributional constraints.
method Gradient flow theory applied to distributionally robust optimization.
result Practical algorithms for sampling from worst-case distributions.
This paper introduces Wasserstein variational inference, a new form of approximate Bayesian inference based on optimal transport theory. Wasserstein variational inference uses a new family of divergences that includes both f-divergences and the Wasserstein distance as special cases. The gradients of the Wasserstein var…
New method optimizes multiple objectives using particle dynamics and gradient flow.
problem Optimizing multiple conflicting objectives in complex scenarios.
method Interacting particle method combining Langevin and birth-death dynamics with a dominance potential.
result Method effectively relocates dominated particles, improving Pareto optimality.
New method improves sampling efficiency in complex stochastic systems.
problem Sampling efficiency in nonconvex stochastic gradient cases.
method Reflection coupling for unadjusted generalized Hamiltonian Monte Carlo.
result Quantitative Gaussian concentration bounds and convergence rates established.
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.
The Wasserstein probability metric has received much attention from the machine learning community. Unlike the Kullback-Leibler divergence, which strictly measures change in probability, the Wasserstein metric reflects the underlying geometry between outcomes. The value of being sensitive to this geometry has been demo…