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.
WAVE improves stability in reinforcement learning by adaptively weighting critic's loss.
problem Inherent instability in actor-critic reinforcement learning algorithms.
method Wasserstein adaptive value estimation with Sinkhorn approximation.
result Achieves $\mathcal{O}\left(\frac{1}{k}
ight)$ convergence rate for critic's mean squared error.
Stability result for a popular algorithm in optimal transport.
problem Stability of the Iterative Proportional Fitting Procedure in time and metric.
method Uniform stability analysis in the 1-Wasserstein metric.
result Quantitative stability result for entropy-regularized Optimal Transport and Schrödinger bridges.
Kim-Milman flow map stable under regular target measures
problem Stability of Kim-Milman flow map under target measure variations
method Stability in relative entropy and 2-Wasserstein distance result Lipschitz stability up to logarithmic factor
Study on reducing dimensionality in high-dimensional regression with kernel methods and stability analysis.
problem Analyzing errors in high-dimensional regression with dimensionality reduction and kernel regression.
method Derive a stability result for kernel regression with Wasserstein distance and apply it to PCA to deduce convergence rates.
result Two-step procedure yields useful convergence rates in semi-supervised settings.
Wasserstein GAN(WGAN) is a model that minimizes the Wasserstein distance between a data distribution and sample distribution. Recent studies have proposed stabilizing the training process for the WGAN and implementing the Lipschitz constraint. In this study, we prove the local stability of optimizing the simple gradien…
We analyze critical points of the Sliced Wasserstein Distance for optimization stability.
problem Understanding the behavior of optimization algorithms for models trained with the Sliced Wasserstein Distance.
method Explicit perturbations and critical point analysis of the SW objective.
result Stable critical points of SW cannot concentrate on segments, providing optimization stability.
Two algorithms estimate Wasserstein distance matrices from few entries for manifold learning.
problem Estimating Wasserstein distance matrices from limited data for manifold learning.
method Proposes two algorithms: matrix completion and Nyström completion for square Wasserstein matrices.
result Nyström completion can outperform matrix completion with a fixed sample budget and improve classification stability.
The paper examines stability of the Sobolev inequality in metric spaces with curvature dimension conditions.
problem Investigating stability of the Sobolev inequality in metric spaces with curvature dimension conditions.
method Assuming almost the same optimal constant, the paper shows that the cumulative distribution of almost extremal functions is close to that of an Aubin-Talenti bubble on the round sphere.
result Quantitative stability with sharp exponent for the Sobolev inequality in various curvature and dimension assumptions.
Improved GAN performance using higher-order Wasserstein moments.
problem Stabilizing and enhancing GANs for better mode coverage and stability.
method Deriving and training a GAN with a modified Wasserstein distance using higher-order moments.
result Training a GAN with higher-order Wasserstein moments improves performance, even with increased computational cost.
Wasserstein distance plays increasingly important roles in machine learning, stochastic programming and image processing. Major efforts have been under way to address its high computational complexity, some leading to approximate or regularized variations such as Sinkhorn distance. However, as we will demonstrate, regu…
This work studies an explicit embedding of the set of probability measures into a Hilbert space, defined using optimal transport maps from a reference probability density. This embedding linearizes to some extent the 2-Wasserstein space, and enables the direct use of generic supervised and unsupervised learning algorit…
Muon dynamics study uses spectral Wasserstein flow for optimization stability.
problem Optimizing deep learning models with gradient normalization.
method Introduces Spectral Wasserstein distances for matrix flows, proving equivalence with Benamou--Brenier formulation.
result Gradient-flow interpretation of mean-field normalized training dynamics.
Unified stability bounds for noisy SGD across convex and non-convex losses.
problem Deriving generalization bounds for noisy stochastic gradient descent.
method Unified approach using Lyapunov functions and applied probability.
result Time-uniform stability bounds for SGD on various loss functions.
Persistence diagrams (PDs) play a key role in topological data analysis (TDA), in which they are routinely used to describe topological properties of complicated shapes. PDs enjoy strong stability properties and have proven their utility in various learning contexts. They do not, however, live in a space naturally endo…
This report has several purposes. First, our report is written to investigate the reproducibility of the submitted paper On the regularization of Wasserstein GANs (2018). Second, among the experiments performed in the submitted paper, five aspects were emphasized and reproduced: learning speed, stability, robustness ag…
Inspired by recent interests of developing machine learning and data mining algorithms on hypergraphs, we investigate in this paper the semi-supervised learning algorithm of propagating "soft labels" (e.g. probability distributions, class membership scores) over hypergraphs, by means of optimal transportation. Borrowin…
Enhances generative models stability and accuracy with BNPL, WMMD, and triple model.
problem Overfitting in GANs and noisy samples in VAEs.
method Bayesian non-parametric learning framework, integrating Wasserstein distance and maximum mean discrepancy.
result Superior performance across various generative tasks.
iWGAN improves GANs by stabilizing training and preventing mode collapse.
problem Stable and effective training of GANs with mode collapse.
method iWGAN combines auto-encoders and WGANs using iterative primal dual optimization.
result iWGAN provides a clear stopping criterion and mitigates mode collapse.
A new portfolio model improves on Kelly's by accounting for estimation error.
problem Estimation error in Kelly portfolio optimization.
method Wasserstein distributionally robust optimization (DRO) to define a robust log-optimal portfolio.
result The Wasserstein-Kelly portfolio outperforms the Kelly portfolio in out-of-sample testing.
A new algorithm computes Wasserstein barycenters without entropic regularization.
problem Computing Wasserstein barycenters efficiently and accurately.
method Free-support algorithm based on particle flow and Riemannian geometry.
result The algorithm avoids entropic regularization and is computationally tractable.
Improved stability for matrix recovery from rank-one measurements.
problem Phase retrieval problem of recovering rank-one positive semidefinite matrices.
method Developed a smoothing Newton method based on Bures-Wasserstein gradient descent.
result Superlinear convergence with rigorous guarantees and stable implementation.
New findings show score matching's accuracy doesn't ensure numerical stability in diffusion sampling.
problem Numerical stability issues in diffusion sampling despite small forward-marginal error.
method Constructing a smooth score field with arbitrarily small forward-marginal L2 error, showing nonexplosive behavior and moments of every order. result Euler--Maruyama discretizations can converge in probability even when moments diverge, demonstrating failure of weak convergence.
Study on stability of optimal transport problems for probability measures.
problem Stability of supermartingale optimal transport problems.
method Approximation in adapted Wasserstein distance and continuity of functional.
result Continuity and monotonicity principles for weak supermartingale optimal transport.
Generative Adversial Networks (GANs) have made a major impact in computer vision and machine learning as generative models. Wasserstein GANs (WGANs) brought Optimal Transport (OT) theory into GANs, by minimizing the 1-Wasserstein distance between model and data distributions as their objective function. Since then, W…
A new GAN loss function based on cumulant generating functions improves stability and robustness.
problem Improving the stability and performance of GANs.
method Cumulant GAN loss function based on variational R{é}nyi divergence.
result Cumulant GAN achieves linear convergence to Nash equilibrium and superior performance in image generation.
New sampling method using regularized Wasserstein proximal for Gibbs distributions.
problem Sampling from Gibbs distributions with numerical stability and efficiency.
method Preconditioned regularized Wasserstein proximal operator.
result Discrete-time convergence analysis and explicit bias characterization.
New bounds for SGLD show error decreases with more data.
problem Establishing generalization error bounds for SGLD in non-convex settings.
method Using dissipativity, smoothness, and uniform stability, time-independent bounds are derived.
result Error bounds decay to zero as sample size increases.
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.
The paper explores theoretical insights into WGANs for better understanding and stability.
problem Stabilizing the training process of GANs.
method Theoretical analysis and statistical convergence study of WGANs.
result Theoretical properties and convergence of WGANs are clarified.
This study investigates self-supervised learning with Wasserstein distance on tree structures.
problem Improving self-supervised learning methods using Wasserstein distance.
method Utilized Tree-Wasserstein distance (TWD) and Jeffrey divergence regularization for training.
result A simple combination of softmax function and Tree-Wasserstein distance outperforms cosine similarity-based methods.
The paper analyzes rates of convergence for optimal transport map estimators using barycentric projections.
problem Estimating optimal transport maps from data sampled according to two distributions.
method Comprehensive analysis of rates of convergence for plug-in estimators defined via barycentric projections.
result New stability estimate for barycentric projections under minimal smoothness assumptions.
Novel stability bounds for OT maps improve density estimation.
problem Estimating optimal transport maps between probability distributions.
method Developed novel stability bounds for OT maps, reducing the problem to density estimation.
result Stability bounds allow for sharper guarantees without smoothness assumptions.
Assume that an agent models a financial asset through a measure Q with the goal to price / hedge some derivative or optimize some expected utility. Even if the model Q is chosen in the most skilful and sophisticated way, she is left with the possibility that Q does not provide an "exact" description of reality. This le…
Introduces new Wasserstein distances for more intrinsic metrics.
problem Improve metric for comparing distributions.
method Introduces RWp distances, designs algorithms for computation. result New distances are more intrinsic and computable.
Batching stabilizes risk in high-dimensional linear regression models.
problem Stability and risk behavior in high-dimensional overparameterized linear regression.
method Minimum-norm overparameterized linear regression model with batch-partitioning.
result Optimal batch size is inversely proportional to noise level and overparametrization ratio, leading to stable risk behavior.
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.
New methods improve stability of Sinkhorn algorithm in machine learning.
problem Stability of Sinkhorn semigroups in high-dimensional settings.
method Semigroup analysis based on contraction coefficients and Lyapunov-type operator-theoretic techniques.
result Unified and simplified arguments in Sinkhorn algorithm stability.
SGD handles label noise with bounds improving over SGLD.
problem Label noise in non-convex optimization.
method Stochastic gradient descent with uniform dissipativity and smoothness conditions, using Wasserstein distance and algorithmic stability.
result Generalization error bounds with a rate of n−2/3, better than SGLD's n−1/2. Deep networks analyzed using IFS theory for stability and generalization.
problem Stability and generalization of deep neural networks.
method Viewing deep architectures as place-dependent IFS and applying results from random dynamical systems.
result Derivation of a Wasserstein generalization bound and a new training objective.
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.
Generative adversarial networks (GANs) are one of the most popular approaches when it comes to training generative models, among which variants of Wasserstein GANs are considered superior to the standard GAN formulation in terms of learning stability and sample quality. However, Wasserstein GANs require the critic to b…
Debiased Wasserstein barycenters improve on entropy regularization in OT.
problem Entropy regularization in OT introduces bias, leading to blurred barycenters.
method Propose debiased Wasserstein barycenters using Sinkhorn iterations.
result Debiased barycenters preserve fast Sinkhorn-like iterations without entropy smoothing bias.
Quantum Earth Mover's distance improves stability and efficiency in quantum learning.
problem Quantum learning's loss landscapes often lead to poor local minima and gradients.
method Introduced the quantum Earth Mover's (EM) distance and proposed a quantum Wasserstein generative adversarial network (qWGAN).
result The quantum EM distance makes quantum learning more stable and efficient.
Proposes a new graph kernel framework using regularized Wasserstein distances.
problem Learning optimal transport distances for graph kernels.
method Introduces Regularized Wasserstein (RW) discrepancy with two regularization terms.
result Empirically validated method outperforms state-of-the-art methods.
Efficiently simulates and calibrates the rough Bergomi model using Wasserstein distance.
problem High computational complexity in pricing and calibration of the rough Bergomi model.
method Developed a modified-sum-of-exponentials Monte Carlo scheme and a calibration approach based on Wasserstein-1 distance.
result The method achieves high pricing accuracy and improved parameter recovery, optimization stability, and out-of-sample performance.
New bounds link SGD's generalization to heavy tails without topological assumptions.
problem Linking SGD's generalization error to heavy tails without additional assumptions.
method Developed Wasserstein stability bounds for heavy-tailed SDEs and their discretizations, converting to generalization bounds.
result Generalization bounds for a broader class of objective functions, including non-convex functions, without topological assumptions.
Work on SGDm under heavy-tailed noise, revealing its generalization properties.
problem Understanding generalization of SGDm under heavy-tailed noise.
method Analysis of continuous-time limit (SDE) and discrete-time SGDm, establishing generalization bounds.
result SGDm can have worse generalization in the presence of heavy-tailed noise for quadratic loss functions.