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Comparing probability distributions is at the crux of many machine learning algorithms. Maximum Mean Discrepancies (MMD) and Wasserstein distances are two classes of distances between probability distributions that have attracted abundant…
We analyze a number of natural estimators for the optimal transport map between two distributions and show that they are minimax optimal. We adopt the plugin approach: our estimators are simply optimal couplings between measures derived…
Optimal transportation, or computing the Wasserstein or ``earth mover's'' distance between two distributions, is a fundamental primitive which arises in many learning and statistical settings. We give an algorithm which solves this problem…
We refer to recent inference methodology and formulate a framework for solving the distributionally robust optimization problem, where the true probability measure is inside a Wasserstein ball around the empirical measure and the radius of…
Robust optimization is a tractable and expressive technique for decision-making under uncertainty, but it can lead to overly conservative decisions when pessimistic assumptions are made on the uncertain parameters. Wasserstein…
In this paper, we develop an exact reformulation and a deterministic approximation for distributionally robust joint chance-constrained programmings (DRCCPs) with a general class of convex uncertain constraints under data-driven Wasserstein…
Multi-marginal optimal transport enables one to compare multiple probability measures, which increasingly finds application in multi-task learning problems. One practical limitation of multi-marginal transport is computational scalability…
We consider a distributionally robust second-order stochastic dominance constrained optimization problem. We require the dominance constraints hold with respect to all probability distributions in a Wasserstein ball centered at the…
Optimal transport provides a metric which quantifies the dissimilarity between probability measures. For measures supported in discrete metric spaces, finding the optimal transport distance has cubic time complexity in the size of the…
The standard quadratic optimization problem (StQP) consists of minimizing a quadratic form over the standard simplex. Without assuming convexity or concavity of the quadratic form, the StQP is NP-hard. This problem has many interesting…
The smooth 1-Wasserstein distance (SWD) $W_1^\sigma$ was recently proposed as a means to mitigate the curse of dimensionality in empirical approximation while preserving the Wasserstein structure. Indeed, SWD exhibits parametric convergence…
While many Machine Learning methods were developed or transposed on Riemannian manifolds to tackle data with known non Euclidean geometry, Optimal Transport (OT) methods on such spaces have not received much attention. The main OT tool on…
Distributionally robust optimization (DRO) has become a powerful framework for estimation under uncertainty, offering strong out-of-sample performance and principled regularization. In this paper, we propose a DRO-based method for linear…
Estimating a $d$-dimensional distribution $\mu$ by the empirical measure $\hat{\mu}_n$ of its samples is an important task in probability theory, statistics and machine learning. It is well known that $\mathbb{E}[\mathcal{W}_p(\hat{\mu}_n,…
This paper presents a novel distribution-agnostic Wasserstein distance-based estimation framework. The goal is to determine an optimal map combining prior estimate with measurement likelihood such that posterior estimation error optimally…
The Wasserstein distances $W_p$ ($p\geq 1$), defined in terms of solution to the Monge-Kantorovich problem, are known to be a useful tool to investigate transport equations. In particular, the Benamou-Brenier formula characterizes the…
Recent years have witnessed a tremendous growth using topological summaries, especially the persistence diagrams (encoding the so-called persistent homology) for analyzing complex shapes. Intuitively, persistent homology maps a potentially…
Covariate shift arises when covariate distributions differ between source and target populations while the conditional distribution of the response remains invariant, and it underlies problems in missing data and causal inference. We…
Discrepancy measures between probability distributions, often termed statistical distances, are ubiquitous in probability theory, statistics and machine learning. To combat the curse of dimensionality when estimating these distances from…
Gaussian mixture models (GMMs) are widely used in machine learning for tasks such as clustering, classification, image reconstruction, and generative modeling. A key challenge in working with GMMs is defining a computationally efficient and…