Note on the computational complexity of Gromov-Wasserstein distance.
problem Computational difficulty of Gromov-Wasserstein distance.
method Analysis of the optimization problem structure and providing explicit examples.
result Gromov-Wasserstein distance optimization problem is non-convex quadratic.
We propose a novel end-to-end non-minimax algorithm for training optimal transport mappings for the quadratic cost (Wasserstein-2 distance). The algorithm uses input convex neural networks and a cycle-consistency regularization to approximate Wasserstein-2 distance. In contrast to popular entropic and quadratic regular…
We propose an SDP relaxation for the Gromov-Wasserstein distance, providing globally optimal solutions.
problem Matching objects between incomparable spaces using the Gromov-Wasserstein distance.
method Semi-definite programming (SDP) relaxation of the GW distance.
result The SDP relaxation provides globally optimal solutions for the GW distance in some instances.
This work characterizes, analytically and numerically, two major effects of the quadratic Wasserstein (W2) distance as the measure of data discrepancy in computational solutions of inverse problems. First, we show, in the infinite-dimensional setup, that the W2 distance has a smoothing effect on the inversion pro…
Recently used in various machine learning contexts, the Gromov-Wasserstein distance (GW) allows for comparing distributions whose supports do not necessarily lie in the same metric space. However, this Optimal Transport (OT) distance requires solving a complex non convex quadratic program which is most of the time very…
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.
New formulations for comparing metric measure spaces with arbitrary positive measures.
problem Comparing metric measure spaces with arbitrary positive measures.
method Two novel formulations: a divergence and a conic lifting approach.
result Efficiently solvable formulations for comparing metric spaces with arbitrary positive measures.
The paper optimizes estimating transport maps between distributions.
problem Estimating optimal transport maps between distributions.
method Plugin approach using optimal couplings and extensions.
result Minimax optimality of the proposed estimators.
The curvature-dimension condition is a generalization of the Bochner inequality to weighted Riemannian manifolds and general metric measure spaces. It is now known to be equivalent to evolution variational inequalities for the heat semigroup, and quadratic Wasserstein distance contraction properties at different times.…
Study shows neural networks trained with GD converge to Gaussian processes with polynomial decay.
problem Understanding convergence of neural networks to Gaussian processes during training.
method Explicit upper bounds on quadratic Wasserstein distance between trained networks and Gaussian approximations.
result Polynomial decay of approximation error with network width and training time.
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.
Bounds neural network output distribution to Gaussian for random initialization.
problem Quantifying the distribution of randomly initialized deep neural networks.
method Quantitative Gaussian approximation using quadratic Wasserstein distance.
result Explicit inequalities show how network sizes affect Gaussian behavior.
New distances measure mixtures of Gaussians, useful in machine learning.
problem Comparing distributions with disjoint supports.
method Schoenberg-Rao distances based on concave Rao's entropy.
result Closed-form distances for mixtures of Gaussians.
SRRM improves recursive transport surrogates in the small-discrepancy regime.
problem Insufficient understanding of recursive partitioning methods' statistical behavior and resolution in the small-discrepancy regime.
method Introduced Selective Recursive Rank Matching (SRRM) to improve the resolution of Recursive Rank Matching (RRM).
result SRRM yields a higher-fidelity practical surrogate for the Wasserstein distance at moderate additional computational cost.
New distances for comparing heterogeneous probability measures efficiently.
problem Comparing probability measures across different spaces.
method Introducing Anchor Energy (AE) and Anchor Wasserstein (AW) distances, and a sweep line algorithm for exact computation.
result Exact computation of AE and AW distances in log-quadratic time, significantly faster than GW.
WEGL embeds graphs in a vector space for faster machine learning.
problem Efficiently embedding graphs for machine learning tasks.
method Wasserstein distance for node embedding similarity, Monge maps for graph representation.
result State-of-the-art classification performance with superior computational efficiency.
Improved efficiency in HMC samplers reduces dissipative behavior.
problem Reducing dissipative behavior in HMC samplers.
method Variable integration time and partial velocity refreshment.
result Efficiency improved by a √κ factor in Wasserstein-2 distance.
Paper solves Gromov-Wasserstein for point clouds efficiently.
problem Quantifying similarity between two formations or shapes.
method Reformulates QAP as low-rank concave quadratic optimization problem.
result Global solution for large-scale problems with thousands of points.
New method improves robust point matching under probabilistic settings.
problem Insufficient theoretical understanding of existing point matching methods.
method Distance profiles and modified matching procedure.
result Improved robustness under probabilistic settings.
Wasserstein GANs fail to approximate Wasserstein distance, leading to their success.
problem Approximating Wasserstein distance in deep generative models.
method Analysis of differences between theoretical setup and training reality.
result Wasserstein GANs' success is due to their failure to approximate Wasserstein distance.
A method for fast estimation of Wasserstein distances using sliced Wasserstein distances.
problem Efficiently computing Wasserstein distances for multiple pairs of distributions.
method Regression on sliced Wasserstein distances to predict true Wasserstein distances.
result The proposed method provides a better approximation of Wasserstein distance than state-of-the-art models, especially in low-data regimes.
The paper introduces a new Wasserstein distance for approximating posteriors in inverse problems.
problem Approximating posterior measures in inverse problems using conditional Wasserstein distances.
method Introduces a conditional Wasserstein distance with restricted couplings and derives its dual.
result Shows that conditional Wasserstein GANs can yield favorable properties for posterior sampling.
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 paper approximates 1-Wasserstein distance using tree-based embedding.
problem Computational inefficiency of estimating 1-Wasserstein distance.
method L1-regularized approach to learn tree weights, using shortest path distance as a linear model.
result Tree-Wasserstein distance (TWD) approximates 1-Wasserstein distance efficiently.
A new slicing method reduces computational cost for cross-domain alignment.
problem High computational cost in solving Gromov-Wasserstein distance.
method Relation-Aware Projecting Direction (RAPD) and Relation-Aware Slicing Distribution (RASD).
result RASGW distance reduces computational cost and improves alignment accuracy.
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.
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 new supervised tree-Wasserstein distance improves document classification.
problem Measuring document similarity efficiently and accurately.
method Rewriting Wasserstein distance on tree metric, using contrastive loss for optimization.
result The Supervised Tree-Wasserstein (STW) distance improves document classification accuracy.
This work robustifies Wasserstein distance estimation with MoM estimators for outlier-polluted data.
problem Estimating Wasserstein distance between two distributions with outliers.
method Introducing MoM-based robust estimators for Wasserstein distance.
result Consistent MoM-based estimators for Wasserstein distance with convergence rates.
Wasserstein distances are increasingly used in a wide variety of applications in machine learning. Sliced Wasserstein distances form an important subclass which may be estimated efficiently through one-dimensional sorting operations. In this paper, we propose a new variant of sliced Wasserstein distance, study the use …
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.
Exact 1-Wasserstein distance between location-scale distributions derived, with privacy effects studied.
problem Calculating the 1-Wasserstein distance between location-scale distributions and its impact on differential privacy.
method Exact expressions and special functions for 1-Wasserstein distance, new upper bounds, and asymptotic analysis.
result New linear upper bound and detailed asymptotic bounds for Gaussian case, effect of differential privacy studied.
New Sliced-Wasserstein distances for non-Euclidean data.
problem Computational burden of Wasserstein distance on non-Euclidean manifolds.
method Derive Sliced-Wasserstein distances and flows on Cartan-Hadamard manifolds.
result General constructions and non-parametric schemes for minimizing new distances.
New tools for estimating and inferring Wasserstein distance in topic models.
problem Estimating and inferring the Wasserstein distance between mixing measures in topic models.
method New canonical interpretation and tools for inference on Wasserstein distance in topic models.
result First minimax lower bounds and fully data-driven inferential tools for the Wasserstein distance in topic models.
The paper explores a new type of kernel using Wasserstein distance for better classification of shapes.
problem Improving kernel methods for shape classification.
method Defined and studied exponential kernels based on regularized Wasserstein distance.
result Wasserstein squared exponential kernels perform better on small shape datasets.
The Sinkhorn "distance", a variant of the Wasserstein distance with entropic regularization, is an increasingly popular tool in machine learning and statistical inference. However, the time and memory requirements of standard algorithms for computing this distance grow quadratically with the size of the data, making th…
The Wasserstein distance and its variations, e.g., the sliced-Wasserstein (SW) distance, have recently drawn attention from the machine learning community. The SW distance, specifically, was shown to have similar properties to the Wasserstein distance, while being much simpler to compute, and is therefore used in vario…
Paper proposes a new Wasserstein distance for mixtures of radially contoured distributions.
problem Generalization of Wasserstein distance to non-elliptically contoured distributions.
method Relaxed formulation for mixtures of radially contoured distributions without marginal consistency.
result The new distance yields more stable error and better color distribution in image transfer tasks.
Introduces spectral-domain Wasserstein distance and Gelbrich bound for elliptical processes.
problem Estimating distances and bounds for elliptical stochastic processes.
method Defines spectral-domain W2 Wasserstein distance and Gelbrich bound. result Develops new spectral-domain bounds for non-elliptical processes.
This paper connects Wasserstein distances to MMD norms for compressive statistical learning.
problem Comparing and controlling Wasserstein distances between probability distributions.
method Establishing conditions under which Wasserstein distances can be controlled by MMD norms.
result Introducing Wasserstein regularity for compressive statistical learning.
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.
Centered plug-in estimators reduce bias in Wasserstein distance estimation.
problem Conservative bias in plug-in estimators of Wasserstein distances.
method Centering procedure based on linear combinations to reduce bias.
result Centered plug-in estimators provide informative upper and lower bounds on Wasserstein distances.
The paper analyzes how the one-dimensional Wasserstein distance captures pointwise density differences in finite samples.
problem Uncertainty in identifying density differences when supports overlap and densities have substantial pointwise differences.
method Analysis using the Poisson process and neural spike train decoding.
result The one-dimensional Wasserstein distance highlights meaningful density differences related to both rate and support.
The paper proves inequalities linking Wasserstein distances and eigenfunctions in RCD(K,∞) spaces.
problem Estimating Wasserstein distances and their bounds in RCD(K,∞) spaces.
method Similar techniques used to prove inequalities involving p-Wasserstein distances and Laplace eigenfunctions. result Proves a conjectured lower bound on p-Wasserstein distance between positive and negative parts of Laplace eigenfunctions. Paper introduces new Gromov-type distances for comparing Gaussian mixture models.
problem Comparing distributions across different metric spaces using Gromov-Wasserstein distances.
method Incorporates invariance properties into MW2, introducing MGW2 and EW2.
result MGW2 and EW2 are efficient for estimating distances between GMMs in practical applications.
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.
We study contractivity properties of gradient flows for functions on normed spaces or, more generally, on Finsler manifolds. Contractivity of the flows turns out to be equivalent to a new notion of convexity for the functions. This is different from the usual convexity along geodesics in non-Riemannian Finsler manifold…
Proposes an efficient lower bound for Gromov-Wasserstein discrepancy.
problem Comparing structured data from different metric-measure spaces.
method Orthogonal Gromov-Wasserstein (OGW) discrepancy with efficient closed-form lower bound.
result Efficient and tight lower bounds for Gromov-Wasserstein discrepancy.