arXiv research

Study uses weak transport for non-convex costs in fixed-income markets.

Study optimal transport on globally hyperbolic spacetimes, focusing on weak Kantorovich potentials' regularity.

We extend martingale transport results to weak martingale transport.

NOT learns optimal transport plans, kernel costs improve performance.

A new method for averaging probability distributions based on optimal weak mass transport.

The paper introduces a new method for risk measurement using weak optimal transport.

New algorithms solve weak optimal transport problems for nonlinear costs.

Under mild regularity assumptions, the transport problem is stable in the following sense: if a sequence of optimal transport plans $π_1, π_2, \ldots$ converges weakly to a transport plan $π$, then $π$ is also optimal (between its marginals). Alfonsi, Corbetta and Jourdain asked whether the same property is true for th…

New emulator bridges simulators using conditional optimal transport.

We study the mean curvature flow with given non-smooth transport term and forcing term, in suitable Sobolev spaces. We prove the global existence of the weak solutions for the mean curvature flow with the terms, by using the modified Allen-Cahn equation that holds useful properties such as the monotonicity formula.

Proves stability in Weyl polytopes using optimal transport.

The study establishes stability in WMOT, crucial for finance with imprecise data.

Extends optimal transport to dynamic and martingale settings.

The paper establishes general results in Lorentzian optimal transport theory.

Study on stability of optimal transport problems for probability measures.

Extends martingale transport for robust finance problems.

It is shown that curvature-dimension bounds CD(N, k) for a metric measure space (X,d,m) in the sense of Sturm imply a weak L^1- Poincare-inequality under some symmetry assumption on the choice of transport rays in the cut locus of (X,d). This condition is satisfied if (X,d) has m-almost surely no branching points.

DPOT uses deep learning to compute optimal transport efficiently.

Study compares synthetic and distributional Ricci curvature bounds.

In this paper we apply techniques from optimal transport to study the neckpinch examples of Angenent-Knopf which arise through the Ricci flow on $\mathbb{S}^{n+1}$. In particular, we recover their proof of 'single-point pinching' along the flow. Using the methods of optimal transportation, we are able to remove the ass…

A mesh-free method solves continuum-marginal optimal transport problems.

Parallel transport defined for 2-bundles over Lie groupoids.

A new algorithm for estimating continuous entropic barycenters under arbitrary costs.

Stein variational gradient descent (SVGD) is a deterministic sampling algorithm that iteratively transports a set of particles to approximate given distributions, based on an efficient gradient-based update that guarantees to optimally decrease the KL divergence within a function space. This paper develops the first th…

Extends martingale Schrödinger bridge to arbitrary dimensions and characterizes it.

A new method for fast optimal transport using sliced Wasserstein generalized geodesics.

Unified Kantorovich duality for multimarginal optimal transport on Polish spaces.

This study compares two methods for sampling with transport maps, finding flow-based proposals work better for multimodal distributions.

A new method selects a representative subsample for efficient kernel density estimation.

Develops optimal transport in Lorentzian spaces with synthetic curvature bounds.

Novel weak solutions for volume-preserving mean curvature flow established.

We deal with irregular curves contained in smooth, closed, and compact surfaces. For curves with finite total intrinsic curvature, a weak notion of parallel transport of tangent vector fields is well-defined in the Sobolev setting. Also, the angle of the parallel transport is a function with bounded variation, and its …

New varifold solutions for mean curvature flow converge and are unique.

Under Markovian assumptions, we leverage a Central Limit Theorem (CLT) for the empirical measure in the test statistic of the composite hypothesis Hoeffding test so as to establish weak convergence results for the test statistic, and, thereby, derive a new estimator for the threshold needed by the test. We first show t…

A general duality proof for Wasserstein distributionally robust optimization.

We propose a framework for solving high-dimensional Bayesian inference problems using \emph{structure-exploiting} low-dimensional transport maps or flows. These maps are confined to a low-dimensional subspace (hence, lazy), and the subspace is identified by minimizing an upper bound on the Kullback--Leibler divergence …

In this work, we extend existing well-posedness by noise results for the stochastic transport and continuity equations by treating them as special cases of the linear advection equation of $k$-forms, which arises naturally in geometric fluid dynamics. In particular, we prove the existence and uniqueness of weak $L^p$-s…

Paper introduces MSA for weakly supervised covariance alignment in MEG signals.

We identify a condition for regularity of optimal transport maps that requires only three derivatives of the cost function, for measures given by densities that are only bounded above and below. This new condition is equivalent to the weak Ma-Trudinger-Wang condition when the cost is $C^4$. Moreover, we only require (n…

Let $A \subset \mathbb{R}^d$, $d\ge 2$, be a compact convex set and let $μ= \varrho_0 dx$ be a probability measure on $A$ equivalent to the restriction of Lebesgue measure. Let $ν= \varrho_1 dx$ be a probability measure on $B_r := \{x\colon |x| \le r\}$ equivalent to the restriction of Lebesgue measure. We prove that t…

We give sufficient conditions for a measured length space (X,d,m) to admit local and global Poincare inequalities. We first introduce a condition DM on (X,d,m), defined in terms of transport of measures. We show that DM, along with a doubling condition on m, implies a scale-invariant local Poincare inequality. We show …

Generative model uses ODEs and RKHSs for measure matching.

Isotonic regression is a standard problem in shape-constrained estimation where the goal is to estimate an unknown nondecreasing regression function $f$ from independent pairs $(x_i, y_i)$ where $\mathbb{E}[y_i]=f(x_i), i=1, \ldots n$. While this problem is well understood both statistically and computationally, much l…

Here we propose a general theoretical method for analyzing the risk bound in the presence of adversaries. Specifically, we try to fit the adversarial learning problem into the minimax framework. We first show that the original adversarial learning problem can be reduced to a minimax statistical learning problem by intr…

New methods estimate transport-growth pairs in unbalanced optimal transport.

We introduce a weak notion of barycenter of a probability measure $μ$ on a metric measure space $(X, d, {\bf m})$, with the metric $d$ and reference measure ${\bf m}$. Under the assumption that optimal transport plans are given by mappings, we prove that our barycenter $B(μ)$ is well defined; it is a probability measur…

By investigating model-independent bounds for exotic options in financial mathematics, a martingale version of the Monge-Kantorovich mass transport problem was introduced in \cite{BeiglbockHenry LaborderePenkner,GalichonHenry-LabordereTouzi}. In this paper, we extend the one-dimensional Brenier's theorem to the present…

Stability of timelike Ricci bounds in low-regularity spacetimes.