The space of Gaussian measures on a Euclidean space is geodesically convex in the -Wasserstein space. This space is a finite dimensional manifold since Gaussian measures are parameterized by means and covariance matrices. By restricting to the space of Gaussian measures inside the -Wasserstein space, we manag…
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48 results for “2-Wasserstein space”
Quantitative stability of optimal transport maps and linearization of the 2-Wasserstein spacestat.ML
Study linearizes 2-Wasserstein space using optimal transport maps.
problem Stability and linearization of the 2-Wasserstein space.
method Explicit embedding of probability measures into a Hilbert space using optimal transport maps.
result The embedding is (bi-)Hölder continuous, with stability results for optimal transport maps.
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.
Study entropic regularization of Gaussian measures and processes on Hilbert space.
problem Regularizing 2-Wasserstein distance for infinite-dimensional Gaussian measures and processes.
method Minimum Mutual Information property, closed form formulas, Fréchet differentiability, Sinkhorn barycenter equation.
result Entropic 2-Wasserstein distance and Sinkhorn divergence are Fréchet differentiable in Hilbert space.
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 work studies Gaussian geometry under entropy-regularized 2-Wasserstein distance.
problem Understanding Gaussian distributions in uncertainty quantification and diffusivity.
method Entropy-regularized 2-Wasserstein distance, closed-form solutions, fixed-point characterization.
result Closed-form expressions for the 2-Sinkhorn divergence and fixed-point barycenter.
We study the heat equation on time-dependent metric measure spaces (as well as the dual and the adjoint heat equation) and prove existence, uniqueness and regularity. Of particular interest are properties which characterize the underlying space as a super Ricci flow as previously introduced by the second author. Our ma…
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.
Embedding complex objects as vectors in low dimensional spaces is a longstanding problem in machine learning. We propose in this work an extension of that approach, which consists in embedding objects as elliptical probability distributions, namely distributions whose densities have elliptical level sets. We endow thes…
Transformers can solve complex filtering problems for non-Gaussian signals.
problem Non-linear and non-Markovian filtering problems for conditionally Gaussian signals.
method Continuous-time transformer models called filterformers.
result Filterformers can approximate the conditional law of non-Markovian and conditionally Gaussian signal processes.
This note shows how independent elliptical distributions minimize the Wasserstein distance.
problem Minimizing the Wasserstein distance between elliptical distributions.
method Analyzing the Wasserstein distance between independent elliptical distributions with the same density generators.
result Independent elliptical distributions minimize their Wasserstein distance from other elliptical distributions with the same density generators.
Introduces spectral-domain Wasserstein distance and Gelbrich bound for elliptical processes.
problem Estimating distances and bounds for elliptical stochastic processes.
method Defines spectral-domain Wasserstein distance and Gelbrich bound.
result Develops new spectral-domain bounds for non-elliptical processes.
New method estimates covariance in deep heteroscedastic regression without labels.
problem Estimating covariance in deep heteroscedastic models is challenging due to sample-dependent covariance and lack of ground truth.
method Proposes a self-supervised approach using KL Divergence and 2-Wasserstein distance for covariance estimation and a neighborhood-based heuristic for pseudo labels.
result Demonstrates effective pseudo labels and a computationally cheaper yet accurate deep heteroscedastic regression.
We prove the equivalence of the curvature-dimension bounds of Lott-Sturm-Villani (via entropy and optimal transport) and of Bakry--Émery (via energy and Γ_2$-calculus) in complete generality for infinitesimally Hilbertian metric measure spaces. In particular, we establish the full Bochner inequality on such metric meas…
Neural Local Wasserstein Regression models distribution-on-distribution regression with flexible, localized transport maps.
problem Estimating distribution-on-distribution regression with global optimal transport maps or linearization limitations.
method Proposes Neural Local Wasserstein Regression, a flexible nonparametric framework using locally defined transport maps in Wasserstein space.
result Demonstrates effective capture of nonlinear and high-dimensional distributional relationships.
New method uses trainable activations to make BNNs behave like GPs.
problem Making Bayesian Neural Networks (BNNs) behave like Gaussian Processes (GPs).
method Introduced trainable activations and periodic activations to map GP priors to BNNs. Used 2-Wasserstein distance for optimization.
result Method consistently outperforms existing approaches or matches heuristic methods.
LightSBB-M improves generative diffusion modeling with lower 2-Wasserstein distances.
problem Improving generative diffusion models using Schrödinger Bridge and Bass methods.
method Optimizes SBB transport plan with dual representation and tunable beta parameter.
result Achieves up to 32% improvement in 2-Wasserstein distance on synthetic datasets.
Speed-accuracy relations for diffusion models: Wisdom from nonequilibrium thermodynamics and optimal transportcond-mat.stat-mech
Diffusion models' speed-accuracy relations derived from thermodynamics.
problem Understanding the trade-off between model speed and accuracy.
method Connecting diffusion models to thermodynamics and optimal transport.
result Speed-accuracy relations derived, providing insights into optimal learning protocols.
New Langevin method achieves third order convergence for strongly log-concave distributions.
problem Sampling from complex distributions efficiently.
method Underdamped Langevin diffusion with third order convergence.
result Achieves 2-Wasserstein error of ε in O(√d/ε^1/3) steps under additional Lipschitz condition.
We prove that on compact Alexandrov spaces with curvature bounded below the gradient flow of the Dirichlet energy in the -space produces the same evolution as the gradient flow of the relative entropy in the -Wasserstein space. This means that the heat flow is well defined by either one of the two gradient fl…
Polynomial networks converge to Gaussian processes at a rate of O(n^(-1/2)).
problem Understanding the convergence rate of polynomial networks to Gaussian processes.
method Examined one-hidden-layer neural networks with random weights, focusing on polynomial activations and their convergence rate in the 2-Wasserstein metric.
result The rate of convergence for polynomial networks to Gaussian processes is $O(n^{- rac{1}{2}})$.
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.
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.
Generative diffusion models improve channel sampling from limited data.
problem Challenges in channel modelling and data collection for wireless systems.
method Diffusion model with U-Net architecture for frequency domain synthesis.
result Stable training and diverse high-fidelity samples generated from true channel distribution.
Extends DAMs to Gaussian distributions for efficient pattern storage and retrieval.
problem Limited storage capacity and retrieval methods for non-vector pattern representations.
method Introduces a log-sum-exp energy function over Gaussian distributions, using optimal transport maps for retrieval dynamics.
result Proves exponential storage capacity and provides quantitative retrieval guarantees.
We present two approaches to the heat flow on a Finsler manifold : either as gradient flow on for the energy; or as gradient flow on the reverse -Wasserstein space of probability measures on for the relative entropy. Both approaches depend on the choice of a measure on …
Develops efficient projections for multivariate probability measures.
problem Estimating causal effects and optimal weights in multivariate data.
method Tangent Wasserstein projections using generalized geodesics.
result Provides a unique solution for causal inference and optimal weights.
Anchored Langevin Algorithmsstat.ML
Unified approach for sampling non-differentiable and heavy-tailed targets.
problem Sampling non-differentiable and heavy-tailed distributions using Langevin algorithms.
method Anchored Langevin dynamics, which modifies the Langevin diffusion with a smooth reference potential and multiplicative scaling.
result Non-asymptotic guarantees in the 2-Wasserstein distance to the target distribution.
Improved reSGLD accelerates convergence in non-convex learning problems.
problem Inefficient swaps due to noisy energy estimators in reSGLD.
method Variance reduction for noisy energy estimators, theoretical analysis, and numerical experiments.
result Exponential acceleration in convergence for non-convex learning problems.
This work extends stochastic localization to joint probability measures for data analysis.
problem Data distributional analysis in high-dimensional probability.
method Unified stochastic localization under Eldan's α-scheme, coupled probability measures via shared Brownian motion.
result Eldan's α-distance as a scalable surrogate for Wasserstein distance.
LMC achieves sqrt(d) dependence in sampling error, improving previous bounds.
problem Analyzing sampling error in Langevin Monte Carlo.
method Refined mean-square analysis for discretizations of contractive SDEs.
result Establishes mixing time bound for LMC.
We characterize lower bounds for the Bakry-Emery Ricci tensor of nonsymmetric diffusion operators by convexity of entropy on the -Wasserstein space, and define a curvature-dimension condition for general metric measure spaces together with a square integrable -form in the sense of \cite{giglinonsmooth}. This ex…
The paper shows how heat flows and Wasserstein distances relate to space rigidity.
problem Understanding rigidity in Wasserstein contraction along heat flows.
method Establishing equivalence between rigidity and Bakry-Émery gradient estimates, applying results from Ambrosio-Brué-Semola and Han.
result Spaces with specific curvature bounds exhibit rigidity in Wasserstein contraction.
We provide a framework to approximate the 2-Wasserstein distance and the optimal transport map, amenable to efficient training as well as statistical and geometric analysis. With the quadratic cost and considering the Kantorovich dual form of the optimal transportation problem, the Brenier theorem states that the optim…
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.
A new sampling method using log-concave Markov chains.
problem Sampling from unnormalized densities efficiently.
method Decomposes sampling into log-concave Markov chains with noisy measurements.
result Shows remarkable capacity to 'tunnel' between modes of a distribution.
FedGTEA learns new tasks in federated learning with task embeddings and alignment.
problem Federated class-incremental learning with task-specific knowledge and model uncertainty.
method Cardinality-Agnostic Task Encoder (CATE) for Gaussian task embeddings, 2-Wasserstein distance for inter-task alignment.
result FedGTEA achieves superior classification performance and mitigates forgetting.
GT is a new method for denoising and enhancing datasets using Gaussian density estimates.
problem Improving latent structures in datasets.
method GT is an iterative method that generates a new distance function by computing the -Wasserstein distance between Gaussian density estimates.
result GT is stable under perturbations and asymptotically ellipsoidal neighborhoods in the continuous case.
New bounds for generative models under weaker assumptions.
problem Establishing convergence guarantees for generative models under weak assumptions.
method Non-asymptotic 2-Wasserstein distance bounds for probability flow ODEs under weak log-concavity and Lipschitz continuity.
result Concrete convergence rates for generative models, including non-log-concave distributions.
Acceleration via silver step-size on Riemannian manifolds with applications to Wasserstein spacemath.OC
New algorithm accelerates optimization on Riemannian manifolds, including Wasserstein space.
problem Accelerating optimization methods in Riemannian geometry.
method Dynamic stepsize algorithms on Riemannian manifolds with specific vector transport.
result First provable accelerated gradient method in Wasserstein space.
The paper studies stability of mean-field variational inference for log-concave distributions.
problem Stability of mean-field variational inference for log-concave distributions.
method Novel approach via linearized optimal transport, lifting non-convex problem to convex optimization over transport maps.
result Dimension-free Lipschitz continuity of the MFVI optimizer with respect to the target distribution, measured in 2-Wasserstein distance.
New algorithm tackles low-rank constraints in optimal transport problems.
problem Optimal transport problems with low-rank constraints.
method Explicit factorization of low-rank couplings as a product of sub-coupling factors linked by a common marginal.
result Stationary convergence of the algorithm proved.
New method aligns diffusion models for inference-time properties without retraining.
problem Aligning pre-trained diffusion models for desired inference-time properties.
method Variationally stable Doob's matching for provable guidance estimation.
result Consistent estimator of guidance with non-asymptotic convergence guarantees.
Discrete time analogues of ergodic stochastic differential equations (SDEs) are one of the most popular and flexible tools for sampling high-dimensional probability measures. Non-asymptotic analysis in the Wasserstein distance of sampling algorithms based on Euler discretisations of SDEs has been recently develop…
Sharp 2-Wasserstein bounds for DDPMs derived from Föllmer process.
problem Sampling error bounds for DDPMs in 2-Wasserstein distance.
method Lipschitz-type conditions on score function, Föllmer process, and log-concave target distributions.
result Sharp upper bounds for DDPMs in 2-Wasserstein distance, optimal in dimension and steps.
The curse of dimensionality affects neural network optimization, especially with smooth functions.
problem The curse of dimensionality in neural network optimization.
method Examined through the evolution of the parameter distribution under 2-Wasserstein gradient flow.
result The curse of dimensionality persists in neural network optimization, even with smooth functions.
The paper studies the geometry of probability measures on the unit circle.
problem Understanding the Riemannian geometry of probability measures on the unit circle.
method Developed an intrinsic framework using the Peter-Weyl Theorem to study the differential geometry of Wasserstein spaces of compact Lie groups.
result Explicitly demonstrated that the Wasserstein space of the unit circle is flat with vanishing curvature.
Study shows uniform-time chaos propagation in mean field Langevin dynamics.
problem Understanding the convergence of marginal distributions in mean field dynamics.
method Assumed functional convexity of energy, used -convergence and Wasserstein metrics.
result Uniform-in-time propagation of chaos proved in both -Wasserstein and relative entropy.