Related papers: Robust Barycenters of Persistence Diagrams
The purpose of this paper is to provide a systematic discussion of a generalized barycenter based on a variant of unbalanced optimal transport (UOT) that defines a distance between general non-negative, finitely supported measures by…
Making sense of Wasserstein distances between discrete measures in high-dimensional settings remains a challenge. Recent work has advocated a two-step approach to improve robustness and facilitate the computation of optimal transport, using…
We develop a general theoretical and algorithmic framework for sparse approximation and structured prediction in $\mathcal{P}_2(\Omega)$ with Wasserstein barycenters. The barycenters are sparse in the sense that they are computed from an…
We develop a general theory to address a consensus-based combination of estimations in a parallelized or distributed estimation setting. Taking into account the possibility of very discrepant estimations, instead of a full consensus we…
We establish novel quantitative stability results for optimal transport problems with respect to perturbations in the target measure. We provide explicit bounds on the stability of optimal transport potentials and maps, which are relevant…
We study the stability of entropically regularized optimal transport with respect to the marginals. Lipschitz continuity of the value and H\"older continuity of the optimal coupling in $p$-Wasserstein distance are obtained under general…
Distributed systems require fusing heterogeneous local probability distributions into a global summary over sparse and unreliable communication networks. Traditional consensus algorithms, which average distributions in Euclidean space,…
In this work we introduce the concept of Bures-Wasserstein barycenter $Q_*$, that is essentially a Fr\'echet mean of some distribution $\mathbb{P}$ supported on a subspace of positive semi-definite Hermitian operators $\mathbb{H}_{+}(d)$.…
The aim of applied topology is to use and develop topological methods for applied mathematics, science and engineering. One of the main tools is persistent homology, an adaptation of classical homology, which assigns a barcode, i.e. a…
We consider probabilistic models for sequential observations which exhibit gradual transitions among a finite number of states. We are particularly motivated by applications such as human activity analysis where observed accelerometer time…
We introduce and study a novel model-selection strategy for Bayesian learning, based on optimal transport, along with its associated predictive posterior law: the Wasserstein population barycenter of the posterior law over models. We first…
In this paper, we focus on computational aspects of the Wasserstein barycenter problem. We propose two algorithms to compute Wasserstein barycenters of $m$ discrete measures of size $n$ with accuracy $\e$. The first algorithm, based on…
The Wasserstein distance, rooted in optimal transport (OT) theory, is a popular discrepancy measure between probability distributions with various applications to statistics and machine learning. Despite their rich structure and…
We introduce a refinement of the persistence diagram, the graded persistence diagram. It is the Mobius inversion of the graded rank function, which is obtained from the rank function using the unary numeral system. Both persistence diagrams…
We are interested in martingale rearrangement couplings. As introduced by Wiesel [37] in order to prove the stability of Martingale Optimal Transport problems, these are projections in adapted Wasserstein distance of couplings between two…
Multi-marginal optimal transport (MOT) is a generalization of optimal transport to multiple marginals. Optimal transport has evolved into an important tool in many machine learning applications, and its multi-marginal extension opens up for…
This paper presents a robust and efficient method for tracking topological features in time-varying scalar data. Structures are tracked based on the optimal matching between persistence diagrams with respect to the Wasserstein metric. This…
A common feature of methods for analyzing samples of probability density functions is that they respect the geometry inherent to the space of densities. Once a metric is specified for this space, the Fr\'echet mean is typically used to…
In this paper, based on the Fr{\'e}chet mean, we define a notion of barycenter corresponding to a usual notion of statistical mean. We prove the existence of Wasserstein barycenters of random distributions defined on a geodesic space (E,…
Optimal transport has recently proved to be a useful tool in various machine learning applications needing comparisons of probability measures. Among these, applications of distributionally robust optimization naturally involve Wasserstein…