Semi-discrete unbalanced optimal transport and quantization

Entropy-regularized optimal transport and its multi-marginal generalization have attracted increasing attention in various applications, in particular due to efficient Sinkhorn-like algorithms for computing optimal transport plans. However, it is often desirable that the marginals of the optimal transport plan do not match the given measures exactly, which led to the introduction of the so-called unbalanced optimal transport. Since unbalanced methods were not examined for the multi-marginal setting so far, we address this topic in the present paper. More precisely, we introduce the unbalanced multi-marginal optimal transport problem and its dual and show that a unique optimal transport plan exists under mild assumptions. Furthermore, we generalize the Sinkhorn algorithm for regularized unbalanced optimal transport to the multi-marginal setting and prove its convergence. For cost functions decoupling according to a tree, the iterates can be computed efficiently. At the end, we discuss three applications of our framework, namely two barycenter problems and a transfer operator approach, where we establish a relation between the barycenter problem and the multi-marginal optimal transport with an appropriate tree-structured cost function.

Unbalanced Optimal Transport, from theory to numerics

Accurate detection of small and dim targets in infrared imagery is a crucial component in infrared search and track which has broad utility in military and remote sensing applications. Low-rank models have enjoyed state-of-the-art performance in infrared tracking applications, but many approaches underutilize dynamics information which has the potential to improve performance in challenging tracking scenarios. We present two algorithms, robust principal components analysis with patched unbalanced optimal transport (RPCA + PUOT) and robust alignment by sparse and low-rank with patched unbalanced optimal transport (RASL + PUOT), which incorporate optimal transport dynamics regularization and demonstrate improved performance on realistic data.

Which is the best metric for the space of collider events? Motivated by the success of the Energy Mover's Distance in characterizing collider events, we explore the larger space of unbalanced optimal transport distances, of which the Energy Mover's Distance is a particular case. Geometric and computational considerations favor an unbalanced optimal transport distance known as the Hellinger-Kantorovich distance, which possesses a Riemannian structure that lends itself to efficient linearization. We develop the particle linearized unbalanced Optimal Transport (pluOT) framework for collider events based on the linearized Hellinger-Kantorovich distance and demonstrate its efficacy in boosted jet tagging. This provides a flexible and computationally efficient optimal transport framework ideally suited for collider physics applications.

Volumetric Boundary Correspondence for Isogeometric Analysis Based on Unbalanced Optimal Transport

This paper introduces a numerical algorithm to compute the L2 optimal transport map between two measures μ and ν, where μ derives from a density ρ defined as a piecewise linear function (supported by a tetrahedral mesh), and where ν is a sum of Dirac masses. I first give an elementary presentation of some known results on optimal transport and then observe a relation with another problem (optimal sampling). This relation gives simple arguments to study the objective functions that characterize both problems. I then propose a practical algorithm to compute the optimal transport map between a piecewise linear density and a sum of Dirac masses in 3D. In this semi-discrete setting [Aurenhammer et al., Proc. of 8th Symposium on Computational Geometry (1992) 350–357] showed that the optimal transport map is determined by the weights of a power diagram. The optimal weights are computed by minimizing a convex objective function with a quasi-Newton method. To evaluate the value and gradient of this objective function, I propose an efficient and robust algorithm, that computes at each iteration the intersection between a power diagram and the tetrahedral mesh that defines the measure μ. The numerical algorithm is experimented and evaluated on several datasets, with up to hundred thousands tetrahedra and one million Dirac masses.

In this paper, we establish a Kantorovich duality for unbalanced optimal total variation transport problems. As consequences, we recover a version of duality formula for partial optimal transports established by Caffarelli and McCann; and we also get another proof of Kantorovich–Rubinstein theorem for generalized Wasserstein distance$\widetilde {W}_1^{a,b}$proved before by Piccoli and Rossi. Then we apply our duality formula to study generalized Wasserstein barycenters. We show the existence of these barycenters for measures with compact supports. Finally, we prove the consistency of our barycenters.

Arbitrary style transfer aims to stylize the content image with the style image. The key problem of style transfer is how to balance the global content structure and the local style patterns. A promising method to solve this problem is the attentional style transfer method, where a learnable embedding of image features enables style patterns to be flexibly recombined with the content image, so local style patterns will be well preserved in the stylized image. However, current attentional style transfer methods cannot well preserve the global content structure. To solve this problem, a novel attentional style transfer network is proposed, that relies on Optimal Transport (OT) for computing the attention map. The proposed OT‐based attention ensures the similarity between global distributions of the synthesized image and its corresponding style image. For the optimal transport computation, a regularized formulation is used, which not only allows an unbalanced optimal transport to address the deviational distributions but also improves the robustness of stylized results. The proposed method finds a well balance between the global content structure and local style patterns. Various experiments are conducted to demonstrate the superiority of the proposed method over state‐of‐the‐art methods.

Optimal transport (OT) has seen a stellar rise in interest and relevance in the past two decades. More recently, severe limitation of OT have started to surface. Two key factors prevent it from becoming a standard tool in general data science applications. The first one is the fact that the era of big data and steadily improving measurement techniques in the natural sciences produce large scale data which is still out of reach for even modern state-of-the-art OT solvers. The second limitation which prevents the reasonable application of OT in several areas is that vanilla OT is only defined between measures of equal total mass intensity (usually probability measures). At the heart of this thesis lies the goal to advance research on OT to allow it to become a standard tool in modern data analysis. To achieve this, this thesis provides contributions to the research on both aforementioned limitations. It provides non-asymptotic deviation bounds for OT barycenters when the underlying measures are estimated from data and uses this to justify a randomised algorithm to approximate OT barycenter while controlling the induced statistical error. Additionally, it considers a specific notion of Unbalanced OT (UOT) and provides a detailled structural and statistical analysis of the resulting (p,C)-Kantorovich-Rubinstein distance and its corresponding barycenters.

Counting dense crowds through computer vision technology has attracted widespread attention. Most crowd counting datasets use point annotations. In this paper, we formulate crowd counting as a measure regression problem to minimize the distance between two measures with different supports and unequal total mass. Specifically, we adopt the unbalanced optimal transport distance, which remains stable under spatial perturbations, to quantify the discrepancy between predicted density maps and point annotations. An efficient optimization algorithm based on the regularized semi-dual formulation of UOT is introduced, which alternatively learns the optimal transportation and optimizes the density regressor. The quantitative and qualitative results illustrate that our method achieves state-of-the-art counting and localization performance.

We introduce and study a simple model capturing the main features of unbalanced optimal transport. It is based on equipping the conical extension of the group of all diffeomorphisms with a natural metric, which allows a Riemannian submersion to the space of volume forms of arbitrary total mass. We describe its finite-dimensional version and present a concise comparison study of the geometry, Hamiltonian features, and geodesics for this and other extensions. One of the corollaries of this approach is that along any geodesic the total mass evolves with constant acceleration, as an object’s height in a constant buoyancy field.

Dynamical formulations of optimal transport (OT) frame the task of comparing distributions as a variational problem which searches for a path between distributions minimizing a kinetic energy functional. In applications, it is frequently natural to require paths of distributions to satisfy additional conditions. Inspired by this, we introduce a model for dynamical OT which incorporates constraints on the space of admissible paths into the framework of unbalanced OT, where the source and target measures are allowed to have a different total mass. Our main results establish, for several general families of constraints, the existence of solutions to the variational problem which defines this path constrained unbalanced OT framework. These results are primarily concerned with distributions defined on an Euclidean space, but we extend them to distributions defined over parallelizable Riemannian manifolds as well. We also consider metric properties of our framework, showing that, for certain types of constraints, our model defines a metric on the relevant space of distributions. This metric is shown to arise as a geodesic distance of a Riemannian metric, obtained through an analogue of Otto’s submersion in the classical OT setting.

This paper defines a new transport metric over the space of nonnegative measures. This metric interpolates between the quadratic Wasserstein and the Fisher–Rao metrics and generalizes optimal transport to measures with different masses. It is defined as a generalization of the dynamical formulation of optimal transport of Benamou and Brenier, by introducing a source term in the continuity equation. The influence of this source term is measured using the Fisher–Rao metric and is averaged with the transportation term. This gives rise to a convex variational problem defining the new metric. Our first contribution is a proof of the existence of geodesics (i.e., solutions to this variational problem). We then show that (generalized) optimal transport and Hellinger metrics are obtained as limiting cases of our metric. Our last theoretical contribution is a proof that geodesics between mixtures of sufficiently close Dirac measures are made of translating mixtures of Dirac masses. Lastly, we propose a numerical scheme making use of first-order proximal splitting methods and we show an application of this new distance to image interpolation.

A relaxation viewpoint to Unbalanced Optimal Transport: Duality, optimality and Monge formulation

The plethora of data from neuroimaging studies provide a rich opportunity to discover effects and generate hypotheses through exploratory data analysis. Brain pathologies often manifest in changes in shape along with deterioration and alteration of brain matter, i.e., changes in mass. We propose a morphometry approach using unbalanced optimal transport that detects and localizes changes in mass and separates them from changes due to the location of mass. The approach generates images of mass allocation and mass transport cost for each subject in the population. Voxelwise correlations with clinical variables highlight regions of mass allocation or mass transfer related to the variables. We demonstrate the method on the white and gray matter segmentations from the OASIS brain MRI data set. The separation of white and gray matter ensures that optimal transport does not transfer mass between different tissues types and separates gray and white matter related changes. The OASIS data set includes subjects ranging from healthy to mild and moderate dementia, and the results corroborate known pathology changes related to dementia that are not discovered with traditional voxel-based morphometry. The transport-based morphometry increases the explanatory power of regression on clinical variables compared to traditional voxel-based morphometry, indicating that transport cost and mass allocation images capture a larger portion of pathology induced changes.