Introduces Wasserstein Transform for updating distance structures.
problem Enhancing features and denoising data sets.
method Representing data points as probability measures and updating distances via Wasserstein distance.
result GT is computationally cheaper and has a closed-form solution for ℓ2-Wasserstein distance. 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.
A new method improves likelihood-free Bayesian inference by transforming summary statistics and using efficient Variational Bayes.
problem Incorrectly assuming normally distributed summary statistics in likelihood-free Bayesian inference.
method Wasserstein Gaussianization transformation combined with robust BSL and efficient Variational Bayes.
result Highly efficient and reliable approximate Bayesian inference for likelihood-free problems.
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.
The study examines lower and upper bounds of Wasserstein distances for affine transformations of random vectors.
problem Understanding Wasserstein distances for affine transformations of random vectors.
method Lower and upper bounds for affine transformations of random vectors in Rn are derived using Bures metric and compositions of affine maps. result Concrete lower bounds and upper bounds for affine transformations are derived and applied to various distributions.
Paper defines new GSW distances for probability measures.
problem Computational simplicity and similarity to Wasserstein distance.
method Generalized Radon transform to define GSW distances.
result GSW and max-GSW distances are distances under certain conditions.
Sparse transformer architecture improves accuracy and speed in generative modeling and inverse problems.
problem Improving accuracy and speed in generative modeling and inverse problems.
method Proposes a sparse transformer architecture using regularized Wasserstein proximal operator with L1 prior. result Sparse transformer achieves higher accuracy and faster convergence than classical methods.
Study geodesic properties of time series data using Wasserstein metric.
problem Modeling nonlinear time series with transport-based metrics.
method Generalized Wasserstein metric and signed cumulative distribution transforms.
result Geodesic properties provide added interpretability and robustness in time series classifiers.
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 Gromov-Wasserstein metric controls rigidity and incorporates prior knowledge.
problem Inflexible Gromov-Wasserstein distance and lack of feature alignment.
method Augmented Gromov-Wasserstein distance with feature alignments and prior knowledge.
result Improved performance in single-cell multi-omic alignment and transfer learning.
Algorithm aligns 3D density maps using Wasserstein distance.
problem Aligning 3D density maps in cryogenic electron microscopy.
method Minimizing 1-Wasserstein distance after rigid transformation using Bayesian optimization.
result Improved accuracy and efficiency in protein molecule alignment.
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 ℓ2-Wasserstein distance between Gaussian density estimates. result GT is stable under perturbations and asymptotically ellipsoidal neighborhoods in the continuous case.
This paper addresses Gaussian Process regression over probability measures, revealing a non-stationarity issue between Euclidean and Wasserstein kernels.
problem Non-stationarity issue between Euclidean and Wasserstein kernels in Gaussian Process regression over probability measures.
method Assuming Euclidean input space, applying algebraic transformation based on uncovered non-stationarity relationship to create a non-stationary and Wasserstein-based Gaussian Process model.
result An algebraic transformation simplifies learning a non-stationary Gaussian Process model over probability measures.
The paper provides convergence guarantees for ODE-based generative models using transformers.
problem Theoretical guarantees for ODE-based generative models.
method A pre-trained autoencoder maps inputs to a latent space, and a transformer predicts the velocity field.
result The distribution of samples generated via estimated ODE flow converges to the target distribution in Wasserstein-2 distance.
GWIL uses Gromov-Wasserstein distance to align expert and imitation agent states.
problem Cross-domain imitation learning challenges due to different system dimensions and stationary distributions.
method Gromov-Wasserstein Imitation Learning (GWIL) using Gromov-Wasserstein distance.
result GWIL effectively aligns expert and imitation agent states in various continuous control domains.
We solve robust optimization problems using Wasserstein balls and apply it to mean-CVaR optimization.
problem Distributionally robust optimization with Wasserstein ambiguity sets.
method Transformed robust optimization into non-robust with penalty term, selecting ambiguity set size.
result Impressive results in robust mean-CVaR optimization compared to other strategies.
This study examines how well GANs estimate the Wasserstein metric.
problem Estimating the Wasserstein metric from samples in GANs.
method Analyzes c-transform formulation to improve Wasserstein metric estimation. result The c-transform does not perform best in the generative setting. 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.
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.
New framework transforms labeled datasets for various machine learning tasks.
problem Lack of principled methods to transform labeled datasets.
method Wasserstein gradient flows in probability space for optimization of data-generating distributions.
result Framework can impose constraints, adapt for transfer learning, or re-purpose models.
A new method for computing shape barycenters from point clouds using Procrustes-Wasserstein distance.
problem Computing representative shapes from point clouds with precise alignment and shape preservation.
method Developed a new distance metric (Procrustes-Wasserstein) and algorithms for computing barycenters.
result Superior performance in precise alignment and shape preservation compared to existing OT approaches.
This paper analyzes convergence of large-scale Transformers with weight decay.
problem Understanding optimization guarantees in large-scale Transformer training.
method Construct mean-field limit, show gradient flow convergence to PDE, demonstrate global minimum consistency.
result Gradient flow reaches global minimum in large-scale Transformers with small weight decay.
The study uncovers invariant features in healthcare models that traditional methods overlook.
problem Discovering overlooked invariant features in healthcare models.
method Empirical learning of transformations minimizing Wasserstein distance and adding similarity regularization.
result LSTM models and BioBERT reveal invariant features not previously recognized.
Modified Wasserstein metric for Gaussian distributions, invariant to isometries.
problem Distance measurement for latent Gaussian distributions invariant to isometries.
method Modified Benamou-Brenier approach leading to a Procrustes Wasserstein metric.
result For Gaussian distributions, the metric reduces to Euclidean distance between eigenvalues.
Paper introduces S3W distance for spherical probability distributions.
problem Comparing spherical probability distributions efficiently and accurately.
method S3W distance using stereographic projection and generalized Radon transform.
result Extensive theoretical analysis and evaluation of S3W performance.
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 paper analyzes deep and wide transformer training dynamics.
problem Understanding the training dynamics of infinitely deep and wide transformers.
method Develops a mean-field framework for gradient-based training of transformers, controlling a neural PDE.
result Establishes a rigorous foundation for gradient-based transformer training, proving convergence to global minima.
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.
Improved estimator reduces bias in statistical learning models.
problem Asymptotic bias in classic WDRO estimator.
method Adjusted Wasserstein distributionally robust estimator.
result Asymptotic unbiased estimator with smaller MSE.
The study improves volatility model pricing accuracy with new statistical expansions.
problem Improving option pricing accuracy in volatility models.
method Developed Edgeworth expansions for various volatility models.
result Enhanced statistical expansions for volatility models.
New gradient flows improve high-dimensional sampling.
problem Sampling from high-dimensional target densities.
method Introducing Radon--Wasserstein gradient flows.
result Linear scaling in particles and dimensions.
LOT embeds distributions for linear separability and classification.
problem Distribution discrimination in various scientific fields.
method Linear Optimal Transport (LOT) embedding into L2 space. result LOT embeds distributions into linearly separable spaces for certain transformations and perturbations.
Paper proposes Sinkformers for Transformers with doubly stochastic attention.
problem Improving Transformer models' accuracy in vision and natural language processing.
method Using Sinkhorn's algorithm to make attention matrices doubly stochastic instead of SoftMax normalization.
result Sinkformers enhance model accuracy in vision and natural language processing tasks.
HW2MP-GAN tackles ancient handwritten text recognition.
problem Automatic text recognition from ancient handwritten records.
method Conditional Generative Adversarial Network (HW2MP-GAN) with Sliced Wasserstein distance and U-Net architectures.
result HW2MP-GAN outperforms state-of-the-art models in image-to-image translation and handwritten recognition.
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.
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.
Study matches two noisy point clouds with geometric transformations and relabeling.
problem Matching two noisy point clouds with orthogonal transformations and relabeling.
method Information-theoretic results and Ping-Pong algorithm for computational alignment.
result The Ping-Pong algorithm retrieves the planted signal after one step.
Efficiently computes optimal transport maps and Wasserstein barycenters using conditional normalizing flows.
problem Computing optimal transport maps and Wasserstein barycenters in high-dimensional spaces.
method Uses conditional normalizing flows to approximate distributions and solve the primal problem.
result Shows computational feasibility for hundreds of input distributions and yields accurate results.
Model approximates continuous functions in 1-Wasserstein space.
problem Approximating continuous functions in 1-Wasserstein space.
method Probabilistic Transformer (PT) model with three phases: feature map, deep neural network, and probabilistic extension of attention mechanism.
result Can approximate any continuous function from R^d to P1(R^D) uniformly on compact sets.
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.
Unified theory of optimal transport for random measures.
problem Statistical uncertainty in optimal transport.
method Constructing L2 over Wasserstein space for random probability measures. result Unified treatment of random optimal transport and principled inference.
Partial Wasserstein Covering aims to identify missing patterns in datasets.
problem Identifying missing patterns in datasets compared to actual applications.
method Formulated as a discrete optimization problem with partial Wasserstein divergence. Proved submodular, allowing greedy approximation. Proposed quasi-greedy algorithms with acceleration techniques.
result Efficiently fills gaps and finds missing scenes in real driving scenes datasets.
New method simulates multivariate extreme events using GANs and Aitchison coordinates.
problem Simulating multivariate extreme events for economic risk assessment.
method Wasserstein-Aitchison GAN approach combining tail dependence and marginal tail modeling.
result Strong performance in capturing tail dependence and generating accurate extreme observations.
The paper develops stochastic methods on geometric spaces for transformations.
problem Existence and uniqueness of stochastic processes on geometric spaces.
method Stochastic parallel transport and equivariant diffusions on the group of diffeomorphisms.
result Existence and uniqueness of stochastic parallel transport and equivariant diffusions.
New method reduces label and data shifts between domains using optimal transport.
problem Label shift between source and target domains in domain adaptation.
method Developed theory and LDROT method to mitigate label and data shifts.
result Theoretical and experimental validation of LDROT's effectiveness.
Regularizes optimal transport for easier computation and stability.
problem Computational challenges in optimal transport problems.
method Semi-dual formulations and entropic regularization.
result Smooth variational problems for simpler implementation and stability.
Transformers can predict new tokens based on any number of context tokens, approximating continuous mappings with fixed resources.
problem Handling an arbitrarily large number of context tokens in transformers.
method Mathematical analysis of transformer's expressivity using Wasserstein distance and continuous mappings.
result Deep transformers are universal and can approximate continuous in-context mappings to arbitrary precision, uniformly over compact token domains.
Study Gaussian-process limits of neural networks using tensor programs.
problem Understanding the behavior of neural networks as they approach infinite width.
method Quantitative analysis through tensor programs and Wasserstein distance.
result Explicit finite-width error bounds, showing convergence to Gaussian-process limits.