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.

Stability result for a popular algorithm in optimal transport.

Model approximates continuous functions in 1-Wasserstein space.

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

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

MPE framework proves universal approximation for quantum data distribution.

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

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

Error estimates found between SGD with momentum and Langevin diffusion.

This work improves scalability of Wasserstein distances 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…

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

Pathfinder uses quasi-Newton optimization for variational inference.

Diffusion models learn multi-modal distributions with optimal efficiency.

pHMC converges on infinite-dimensional spaces with bounds.

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…

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

Algorithm aligns 3D density maps using Wasserstein distance.

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…

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

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

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…

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

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…

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

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…

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 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…

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.

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

The paper analyzes reflected diffusion models on hypercube data.

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

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…

CDM models counterfactual outcomes in longitudinal data with improved accuracy.

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

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

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…

In Federated Learning, we aim to train models across multiple computing units (users), while users can only communicate with a common central server, without exchanging their data samples. This mechanism exploits the computational power of all users and allows users to obtain a richer model as their models are trained …

New error bounds for GANs with nonlinear objective functions derived.

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

New proof shows incremental flow models are essential for universal generation.

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…