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.
We propose fast approximations for the generalized sliced-Wasserstein distance.
problem Efficient approximation of the generalized sliced-Wasserstein distance in high dimensions.
method Deterministic approximations using random projections and concentration of measure results.
result One-dimensional projections of high-dimensional random vectors are approximately Gaussian.
A faster Wasserstein k-means algorithm for histogram data reduces computation and maintains clustering quality.
problem Efficiently clustering histogram data with reduced computation time.
method Sparse simplex projection to reduce data samples, centroids, and ground cost matrix, dynamically removing lower-valued samples.
result Significant reduction in computational complexity without compromising clustering quality.
Paper develops a new method for differential privacy sampling using Wasserstein distance.
problem Sampling from distributions under differential privacy constraints with geometric structure consideration.
method Develops a novel framework with Wasserstein Projection Mechanism (WPM) for minimax optimal mechanisms.
result Proposes efficient algorithms for approximate computation of the Wasserstein Projection Mechanism.
Improves point-cloud reconstruction by optimizing projections with self-attention.
problem Inefficient and non-metric projection methods for sliced Wasserstein distances.
method Proposes distributional sliced Wasserstein distance with self-attention for permutation-invariant and metric optimization.
result Self-attention amortized distributional projection optimization achieves better performance in point-cloud reconstruction.
A new method optimizes projection directions for sliced Wasserstein distances.
problem Finding informative projecting directions for sliced Wasserstein distances is computationally expensive.
method Amortized projection optimization to predict directions efficiently.
result Proposed amortized models improve generative modeling performance.
A new distance measure balances projection exploration and informativeness.
problem Inefficient and incomplete projection sampling in existing sliced-Wasserstein distances.
method Proposes Distributional Sliced-Wasserstein (DSW) that optimally balances projection exploration and informativeness.
result DSW generalizes Max-SW and can be computed efficiently.
Paper presents efficient computation of robust Wasserstein distance using Riemannian optimization.
problem Intractability of optimizing Projection Robust Wasserstein (PRW) distance due to non-convexity and non-smoothness.
method Riemannian optimization to efficiently compute PRW/Wasserstein Projection Pursuit (WPP) distance.
result The original formulation of PRW/WPP can be efficiently computed in practice, providing better behavior than its convex relaxation.
Paper proposes PRWB and RPRWB for Wasserstein barycenters.
problem Numerical challenges in computing Wasserstein barycenters.
method Projection robust Wasserstein barycenter (PRWB) and relaxed PRWB (RPRWB).
result RPRWB improves clustering performance on real text datasets.
Researchers find Busemann functions in Wasserstein space, enabling efficient projections and distances.
problem Defining Busemann functions in Wasserstein space for efficient data projections and distances.
method Investigated existence and computation of Busemann functions in Wasserstein space, establishing closed-form expressions for specific cases.
result Explicit projection schemes for probability distributions on \(\mathbb{R}\) enable novel Sliced-Wasserstein distances over Gaussian mixtures and labeled datasets.
New method improves Wasserstein distance for large-scale data.
problem High computational cost of Wasserstein distance for large-scale machine learning.
method Augmented Sliced Wasserstein Distances (ASWDs) using neural network mappings.
result ASWDs significantly outperform other Wasserstein variants in synthetic and real-world problems.
A new slicing method speeds up sliced Wasserstein estimation.
problem Efficiently estimating sliced Wasserstein distance.
method Random-Path Projecting Direction (RPD) for fast sampling.
result RPSW and IWRPSW show favorable performance in training generative models.
A new method approximates the Sliced-Wasserstein distance without random projections.
problem Efficiently approximating the Sliced-Wasserstein distance for machine learning applications.
method Utilizing the concentration of measure phenomenon to develop a deterministic approximation.
result The approximation error goes to zero as the dimension increases, under a weak dependence condition.
Introduces MSW distances to improve SW metrics.
problem Redundant projections in SW distance.
method Imposes Markov structure on projecting directions.
result MSW distances improve SW metrics.
The paper finds non-Gaussian directions in high-dimensional data using Wasserstein distance.
problem Locating interesting non-Gaussian features in high-dimensional data.
method Projection pursuit using 2-Wasserstein distance to maximize the difference from Gaussian.
result Statistical guarantees for accurately approximating an unknown low-dimensional non-Gaussian subspace.
This work improves understanding of projection robust optimal transport distances.
problem Understanding the behavior of minimum Wasserstein estimators in high-dimensional and misspecified models.
method Adopting projection robust (PR) optimal transport, establishing statistical properties, proposing IPRW distance, and providing asymptotic guarantees.
result Established fundamental statistical properties and proposed new distances that outperform Wasserstein distances empirically.
New method estimates SW distance using CDFs for scalable data parallelism.
problem Estimating SW distance efficiently for large datasets.
method Estimators based on CDFs of projected measures, avoiding sorting.
result Efficient estimation for large datasets and federated learning.
New statistical methods for analyzing distributions using Wasserstein metric.
problem Statistical analysis of probability distributions on the real line.
method Projected methods exploiting Wasserstein metric and Riemannian structure.
result Projected PCA and regression methods are faster and more flexible.
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.
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.
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.
Our attacks are stronger and faster under Wasserstein metric.
problem Vulnerability of deep models to adversarial attacks.
method Developed an exact yet efficient projection operator and used the Frank-Wolfe method.
result Generated much stronger attacks and improved model robustness.
A new method for comparing image probability measures using convolution operators.
problem Efficiently comparing images using conventional sliced Wasserstein methods.
method Proposed convolution sliced Wasserstein (CSW) methods with stride, dilation, and non-linear activation.
result CSW demonstrates favorable performance over conventional sliced Wasserstein in image comparison and deep generative modeling.
A new method solves the projection robust Wasserstein distance problem efficiently.
problem Computing the projection robust Wasserstein distance is challenging due to the curse of dimensionality.
method Riemannian block coordinate descent (RBCD) method to solve the regularized max-min problem over the Stiefel manifold.
result RBCD method significantly improves the complexity of obtaining an ε-stationary point compared to existing methods.
Proposes variance reduction techniques for sliced Wasserstein distance estimation.
problem Intractability of estimating sliced Wasserstein distances.
method Uses control variates based on Gaussian approximations of projected measures.
result Significant reduction in variance of SW distance estimators.
A general duality proof for Wasserstein distributionally robust optimization.
problem Optimizing under uncertainty with Wasserstein distance.
method One-dimensional convex analysis and interchangeability principle.
result General duality result holds for various distributions and costs.
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.
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.
New hyperbolic sliced-Wasserstein distances derived for efficient comparison.
problem Efficient comparison of distributions in hyperbolic spaces.
method Projections on geodesics or horospheres to derive novel sliced-Wasserstein distances.
result Novel hyperbolic sliced-Wasserstein distances are more computationally efficient.
Generative adversarial nets (GANs) and variational auto-encoders have significantly improved our distribution modeling capabilities, showing promise for dataset augmentation, image-to-image translation and feature learning. However, to model high-dimensional distributions, sequential training and stacked architectures …
In this paper, for μ and ν two probability measures on Rd with finite moments of order ρ≥1, we define the respective projections for the Wρ-Wasserstein distance of μ and ν on the sets of probability measures dominated by ν and of probability measures larger than μ in the convex order. Th…
Wasserstein Discriminant Analysis (WDA) is a new supervised method that can improve classification of high-dimensional data by computing a suitable linear map onto a lower dimensional subspace. Following the blueprint of classical Linear Discriminant Analysis (LDA), WDA selects the projection matrix that maximizes the …
New method uses spherical convolutional Wasserstein distance to validate climate models.
problem Ensuring the accuracy of global climate models.
method Spherical convolutional Wasserstein distance to measure model differences.
result Phase 6 models show modest improvements in realistic climatologies.
A new metric HCP distance for comparing distributions.
problem Comparing high-dimensional probability distributions efficiently.
method Hilbert curve projection to low-dimensional coupling, followed by transport distance calculation.
result HCP distance is a proper metric for probability measures with bounded supports.
A new approach simplifies Sliced-Wasserstein distances to improve learning performance.
problem The concentration of measure phenomenon makes random projections uninformative in high dimensions.
method Propose rescaling the 1D Wasserstein distance to make all slices equally informative.
result The classical Sliced-Wasserstein, properly configured, can match or surpass complex variants.
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.
Proposes PER loss to regularize neural network activations to normal distribution.
problem Improving neural network generalization and training speed.
method Regularizes activations to standard normal distribution via projected error function and Wasserstein distance.
result Minimizes Wasserstein distance between activation distribution and standard normal.
Efficient algorithms solve large-scale DRSVM problems.
problem Optimizing support vector machines under worst-case distribution uncertainty.
method Epigraphical projection-based incremental algorithms.
result Incremental algorithms solve DRSVM problems up to 1000x faster than state-of-the-art methods.
This paper connects nonpositive sectional curvature of a Riemannian manifold with the displacement convexity of the variance functional on the space P(M) of probability measures over M. We show that M has nonpositive sectional curvature and has trivial topology (i.e, is homeomorphic to Rn) if and only…
A rapidly growing area of work has studied the existence of adversarial examples, datapoints which have been perturbed to fool a classifier, but the vast majority of these works have focused primarily on threat models defined by ℓp norm-bounded perturbations. In this paper, we propose a new threat model for adver…
A new metric for comparing measures on tree systems reduces computational burden.
problem Heavy computation in Optimal Transport problems.
method Introducing tree systems and a novel metric (Tree-Sliced Wasserstein distance on Systems of Lines, TSW-SL).
result TSW-SL performs favorably compared to Sliced Wasserstein and its variants.
New algorithm improves signal recovery from noisy measurements with theoretical guarantees.
problem Recovering signals from noisy measurements in inverse problems.
method Wasserstein-based projections (WP) replacing analytic regularization with data-driven denoising.
result WP approximates true projection with high probability, providing theoretical guarantees.
Bayesian approach to portfolio selection reduces pessimism in frequent trading.
problem Tackling the challenge of estimating drift in Merton's portfolio selection model.
method Bayesian distributionally robust control with nonlinear Wasserstein projections.
result Reduced pessimism and improved performance in frequent rebalancing compared to existing methods.
The study examines the distribution of projections of Gaussian data points and its implications for learning models.
problem Understanding the distribution of projections of Gaussian data points in high dimensions.
method Analyzes the asymptotic behavior of projections of i.i.d. standard Gaussian vectors in Rd onto m-dimensional subspaces. result Establishes bounds on the Wasserstein radius of the set of probability distributions arising from these projections.
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.
High-dimensional models trained on smooth manifolds achieve optimal rates in Wasserstein metrics.
problem Training score-based generative models on complex, low-dimensional manifolds.
method Proves optimal rates for SGMs on smooth manifolds, separating into noise regimes and using ReLU nearest-projection coordinates.
result Optimal intrinsic Wasserstein rates are achieved, with polynomial ambient dependence for families with controlled geometry and density.
Paper proposes PPMM for fast estimation of large-scale OTM.
problem Estimation of large-scale optimal transport maps (OTM) is challenging due to the curse of dimensionality.
method Combines projection pursuit regression and sufficient dimension reduction to adaptively select projection directions.
result PPMM consistently estimates the most informative projection direction and weakly converges to the target OTM.
Robust 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