Semi-discrete unbalanced optimal transport and quantization – ERRATUM

In this paper we study the class of optimal entropy-transport problems introduced by Liero, Mielke and Savaré in Inventiones Mathematicae 211 in 2018. This class of unbalanced transport metrics allows for transport between measures of different total mass, unlike classical optimal transport where both measures must have the same total mass. In particular, we develop the theory for the important subclass of semi-discrete unbalanced transport problems, where one of the measures is diffuse (absolutely continuous with respect to the Lebesgue measure) and the other is discrete (a sum of Dirac masses). We characterize the optimal solutions and show they can be written in terms of generalized Laguerre diagrams. We use this to develop an efficient method for solving the semi-discrete unbalanced transport problem numerically. As an application, we study the unbalanced quantization problem, where one looks for the best approximation of a diffuse measure by a discrete measure with respect to an unbalanced transport metric. We prove a type of crystallization result in two dimensions – optimality of a locally triangular lattice with spatially varying density – and compute the asymptotic quantization error as the number of Dirac masses tends to infinity.

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.

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.

This paper explores a novel connection between two areas: shape analysis of surfaces and unbalanced optimal transport. Specifically, we characterize the square root normal field (SRNF) shape distance as the pullback of the Wasserstein–Fisher–Rao (WFR) unbalanced optimal transport distance. In addition we propose a new algorithm for computing the WFR distance and present numerical results that highlight the effectiveness of this algorithm. As a consequence of our results we obtain a precise method for computing the SRNF shape distance directly on piecewise linear surfaces and gain new insights about the degeneracy of this distance.

This contribution presents substantial computational advancements to compare measures even with varying masses. Specifically, we utilize the nonequispaced fast Fourier transform to accelerate the radial kernel convolution in unbalanced optimal transport approximation, built upon the Sinkhorn algorithm. We also present accelerated schemes for maximum mean discrepancies involving kernels. Our approaches reduce the arithmetic operations needed to compute distances from On2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\mathcal {O}}}\\left( n^{2}\\right) $$\\end{document} to Onlogn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{{\\mathcal {O}}}}\\left( n \\log n \\right) $$\\end{document}, opening the door to handle large and high-dimensional datasets efficiently. Furthermore, we establish robust connections between transportation problems, encompassing Wasserstein distance and unbalanced optimal transport, and maximum mean discrepancies. This empowers practitioners with compelling rationale to opt for adaptable distances.

Multi-view multi-object tracking algorithms are expected to resolve multi-object tracking persistent issues within a single camera. However, the inconsistency of camera videos in most of the surveillance systems obstructs the ability of re-identifying and jointly tracking targets through different views. As a crucial task in multi-camera tracking, assigning targets from one view to another is considered as an assignment problem. This paper is presenting an alternative approach based on Unbalanced Optimal Transport for the unbalanced assignment problem. On each view, targets' position and appearance are projected on a learned metric space, and then an Unbalanced Optimal Transport algorithm is applied to find the optimal assignment of targets between pairs of views. The experiments on common multi-camera databases show the superiority of our proposal to the heuristic approach on MOT metrics.

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.

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.

Unbalanced Optimal Transport, from theory to numerics

Volumetric Boundary Correspondence for Isogeometric Analysis Based on Unbalanced Optimal Transport

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.

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.

SUMMARY Full waveform inversion (FWI) is an important and popular technique in subsurface Earth property estimation. In this paper, several improvements to the FWI methodology are developed and demonstrated with numerical examples, including a simple two-layer seismic velocity model, a cross borehole Camembert model and a surface seismic Marmousi model. We introduce an unbalanced optimal transport (UOT) distance with Kullback–Leibler divergence to replace the L2 distance in the FWI problem. Also, a mixed L1/Wasserstein distance is constructed that preserves the convex properties with respect to shift, dilation, and amplitude change operation. An entropy regularization approach and convolutional scaling algorithms are used to compute the distance and the gradient efficiently. Two strategies of normalization methods that transform the seismic signals into non-negative functions are discussed. The numerical examples are then presented at the end of the paper.

Radiomic clustering using graph network techniques coupled with unbalanced optimal transport