arXiv research

Exact 1-Wasserstein distance between location-scale distributions derived, with privacy effects studied.

This paper approximates 1-Wasserstein distance using tree-based embedding.

WGANs use optimal 1-Wasserstein distance to generate distributions.

Deep neural networks can approximate any target probability distribution given certain conditions.

This work connects Cramér distance to QR-DQN for DRL.

This work studies the smooth 1-Wasserstein distance and its limit distribution in high dimensions.

Learning algorithms for implicit generative models can optimize a variety of criteria that measure how the data distribution differs from the implicit model distribution, including the Wasserstein distance, the Energy distance, and the Maximum Mean Discrepancy criterion. A careful look at the geometries induced by thes…

This work improves scalability of Wasserstein distances in high dimensions.

Error estimates found between SGD with momentum and Langevin diffusion.

A new robust metric compares distributions more accurately than existing methods.

Algorithm aligns 3D density maps using Wasserstein distance.

MPE framework proves universal approximation for quantum data distribution.

Sharp bounds for max-sliced Wasserstein distances derived for empirical distributions.

LSDM uses unpaired data to match latent space distributions for generative modeling.

Pathfinder uses quasi-Newton optimization for variational inference.

pHMC converges on infinite-dimensional spaces with bounds.

Diffusion models learn multi-modal distributions with optimal efficiency.

Paper analyzes statistical efficiency of TD learning in Hilbert spaces with Freedman's inequality.

This paper improves topic model estimation for sparse distributions and applies it to Wasserstein distances.

Combinatorial approach to $α$-Ricci and Lin-Lu-Yau Ricci curvatures on graphs

The paper explores multidimensional critic output in GANs, improving convergence and diversity.

We prove that for combinatorial graphs with non-negative Ollivier curvature, one has \[ \|P_t μ- P_t ν\|_1 \leq \frac{W_1(μ,ν)}{\sqrt{t}} \] for all probability measures $μ,ν$ where $P_t$ is the heat semigroup and $W_1$ is the $\ell_1$-Wasserstein distance. This turns out to be an equivalent formulation of a version of…

We consider the problem of sampling from a target distribution, which is \emph {not necessarily logconcave}, in the context of empirical risk minimization and stochastic optimization as presented in Raginsky et al. (2017). Non-asymptotic analysis results are established in the $L^1$-Wasserstein distance for the behavio…

Stability result for a popular algorithm in optimal transport.

This study investigates self-supervised learning with Wasserstein distance on tree structures.

Topological data analysis offers a rich source of valuable information to study vision problems. Yet, so far we lack a theoretically sound connection to popular kernel-based learning techniques, such as kernel SVMs or kernel PCA. In this work, we establish such a connection by designing a multi-scale kernel for persist…

Develops a new divergence framework that combines $f$-divergences and IPMs.

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…

Recent methods for generating novel molecules use graph representations of molecules and employ various forms of graph convolutional neural networks for inference. However, training requires solving an expensive graph isomorphism problem, which previous approaches do not address or solve only approximately. In this wor…

We study unsupervised generative modeling in terms of the optimal transport (OT) problem between true (but unknown) data distribution $P_X$ and the latent variable model distribution $P_G$. We show that the OT problem can be equivalently written in terms of probabilistic encoders, which are constrained to match the pos…

Many attempts have been made in recent decades to integrate machine learning (ML) and topological data analysis. A prominent problem in applying persistent homology to ML tasks is finding a vector representation of a persistence diagram (PD), which is a summary diagram for representing topological features. From the pe…

Model approximates continuous functions in 1-Wasserstein space.

The paper analyzes reflected diffusion models on hypercube data.

This thesis explores Ollivier-Ricci curvature in graphs and manifolds, with applications to graph neural networks.

Particle-optimization-based sampling (POS) is a recently developed effective sampling technique that interactively updates a set of particles. A representative algorithm is the Stein variational gradient descent (SVGD). We prove, under certain conditions, SVGD experiences a theoretical pitfall, {\it i.e.}, particles te…

We study the problem of sampling from a distribution $p^*(x) \propto \exp\left(-U(x)\right)$, where the function $U$ is $L$-smooth everywhere and $m$-strongly convex outside a ball of radius $R$, but potentially nonconvex inside this ball. We study both overdamped and underdamped Langevin MCMC and establish upper bound…

New method learns chaotic dynamics from single noisy trajectory.

Diverse projection ensembles improve distributional reinforcement learning.

CDM models counterfactual outcomes in longitudinal data with improved accuracy.

The paper tackles learning from non-irreducible Markov chains, proving learnability and generalization bounds.

Personalized federated learning adapts models to each user's data.

Momentum methods such as Polyak's heavy ball (HB) method, Nesterov's accelerated gradient (AG) as well as accelerated projected gradient (APG) method have been commonly used in machine learning practice, but their performance is quite sensitive to noise in the gradients. We study these methods under a first-order stoch…

Persistent homology (PH) is a rigorous mathematical theory that provides a robust descriptor of data in the form of persistence diagrams (PDs) which are 2D multisets of points. Their variable size makes them, however, difficult to combine with typical machine learning workflows. In this paper we introduce persistence c…

Improved persistence spheres map measures to functions, stable under partial transport.

New method for high-dimensional linear regression using empirical Bayes.

STRAND: A single representation for hypothesis testing and vectorisation of persistence diagrams

Given a sample $Y$ from an unknown manifold $X$ embedded in Euclidean space, it is possible to recover the homology groups of $X$ by building a Vietoris--Rips or Čech simplicial complex on top of the vertex set $Y$. However, these simplicial complexes need not inherit the metric structure of the manifold, in particular…

The paper tightens bounds on distances between Reeb graphs.