Optimal Transport Approach Research Articles

Sharp Sobolev inequalities on noncompact Riemannian manifolds with Ric≥0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extsf{Ric}\\ge 0$$\\end{document} via optimal transport theory

In their seminal work, Cordero-Erausquin, Nazaret and Villani (Adv Math 182(2):307-332, 2004) proved sharp Sobolev inequalities in Euclidean spaces via Optimal Transport, raising the question whether their approach is powerful enough to produce sharp Sobolev inequalities also on Riemannian manifolds. By using L1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^1$$\\end{document}-optimal transport approach, the compact case has been successfully treated by Cavalletti and Mondino (Geom Topol 21:603-645, 2017), even on metric measure spaces verifying the synthetic lower Ricci curvature bound. In the present paper we affirmatively answer the above question for noncompact Riemannian manifolds with non-negative Ricci curvature; namely, by using Optimal Transport theory with quadratic distance cost, sharp Lp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^p$$\\end{document}-Sobolev and Lp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^p$$\\end{document}-logarithmic Sobolev inequalities (both for p>1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p>1$$\\end{document} and p=1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p=1$$\\end{document}) are established, where the sharp constants contain the asymptotic volume ratio arising from precise asymptotic properties of the Talentian and Gaussian bubbles, respectively. As a byproduct, we give an alternative, elementary proof to the main result of do Carmo and Xia (Math 140:818-826, 2004) and subsequent results, concerning the quantitative volume non-collapsing estimates on Riemannian manifolds with non-negative Ricci curvature that support Sobolev inequalities.

An optimal transport approach to estimating causal effects via nonlinear difference-in-differences

Abstract We propose a nonlinear difference-in-differences (DiD) method to estimate multivariate counterfactual distributions in classical treatment and control study designs with observational data. Our approach sheds a new light on existing approaches like the changes-in-changes estimator and the classical semiparametric DiD estimator, and it also generalizes them to settings with multivariate heterogeneity in the outcomes. The main benefit of this extension is that it allows for arbitrary dependence between the coordinates of vector potential outcomes and includes higher-dimensional unobservables, something that existing methods cannot provide in general. We demonstrate its utility on both synthetic and real data. In particular, we revisit the classical Card & Krueger dataset, which reports fast food restaurant employment before and after a minimum wage increase. A reanalysis with our methodology suggests that these restaurants substitute full-time labor with part-time labor on aggregate in response to a minimum wage increase. This treatment effect requires estimation of the multivariate counterfactual distribution, an object beyond the scope of classical causal estimators previously applied to this data.

Constrained Reweighting of Distributions: An Optimal Transport Approach.

We commonly encounter the problem of identifying an optimally weight-adjusted version of the empirical distribution of observed data, adhering to predefined constraints on the weights. Such constraints often manifest as restrictions on the moments, tail behavior, shapes, number of modes, etc., of the resulting weight-adjusted empirical distribution. In this article, we substantially enhance the flexibility of such a methodology by introducing a nonparametrically imbued distributional constraint on the weights and developing a general framework leveraging the maximum entropy principle and tools from optimal transport. The key idea is to ensure that the maximum entropy weight-adjusted empirical distribution of the observed data is close to a pre-specified probability distribution in terms of the optimal transport metric, while allowing for subtle departures. The proposed scheme for the re-weighting of observations subject to constraints is reminiscent of the empirical likelihood and related ideas, but offers greater flexibility in applications where parametric distribution-guided constraints arise naturally. The versatility of the proposed framework is demonstrated in the context of three disparate applications where data re-weighting is warranted to satisfy side constraints on the optimization problem at the heart of the statistical task-namely, portfolio allocation, semi-parametric inference for complex surveys, and ensuring algorithmic fairness in machine learning algorithms.

Unbalanced regularized optimal mass transport with applications to fluid flows in the brain

As a generalization of the optimal mass transport (OMT) approach of Benamou and Brenier’s, the regularized optimal mass transport (rOMT) formulates a transport problem from an initial mass configuration to another with the optimality defined by the total kinetic energy, but subject to an advection-diffusion constraint equation. Both rOMT and the Benamou and Brenier’s formulation require the total initial and final masses to be equal; mass is preserved during the entire transport process. However, for many applications, e.g., in dynamic image tracking, this constraint is rarely if ever satisfied. Therefore, we propose to employ an unbalanced version of rOMT to remove this constraint together with a detailed numerical solution procedure and applications to analyzing fluid flows in the brain.

Cellstitch: 3D cellular anisotropic image segmentation via optimal transport

BackgroundSpatial mapping of transcriptional states provides valuable biological insights into cellular functions and interactions in the context of the tissue. Accurate 3D cell segmentation is a critical step in the analysis of this data towards understanding diseases and normal development in situ. Current approaches designed to automate 3D segmentation include stitching masks along one dimension, training a 3D neural network architecture from scratch, and reconstructing a 3D volume from 2D segmentations on all dimensions. However, the applicability of existing methods is hampered by inaccurate segmentations along the non-stitching dimensions, the lack of high-quality diverse 3D training data, and inhomogeneity of image resolution along orthogonal directions due to acquisition constraints; as a result, they have not been widely used in practice.MethodsTo address these challenges, we formulate the problem of finding cell correspondence across layers with a novel optimal transport (OT) approach. We propose CellStitch, a flexible pipeline that segments cells from 3D images without requiring large amounts of 3D training data. We further extend our method to interpolate internal slices from highly anisotropic cell images to recover isotropic cell morphology.ResultsWe evaluated the performance of CellStitch through eight 3D plant microscopic datasets with diverse anisotropic levels and cell shapes. CellStitch substantially outperforms the state-of-the art methods on anisotropic images, and achieves comparable segmentation quality against competing methods in isotropic setting. We benchmarked and reported 3D segmentation results of all the methods with instance-level precision, recall and average precision (AP) metrics.ConclusionsThe proposed OT-based 3D segmentation pipeline outperformed the existing state-of-the-art methods on different datasets with nonzero anisotropy, providing high fidelity recovery of 3D cell morphology from microscopic images.

Optimal transport approach to Michael–Simon–Sobolev inequalities in manifolds with intermediate Ricci curvature lower bounds

Optimal transport approach to Michael–Simon–Sobolev inequalities in manifolds with intermediate Ricci curvature lower bounds

A Graph-Space Optimal Transport Approach Based on Kaniadakis κ-Gaussian Distribution for Inverse Problems Related to Wave Propagation.

Data-centric inverse problems are a process of inferring physical attributes from indirect measurements. Full-waveform inversion (FWI) is a non-linear inverse problem that attempts to obtain a quantitative physical model by comparing the wave equation solution with observed data, optimizing an objective function. However, the FWI is strenuously dependent on a robust objective function, especially for dealing with cycle-skipping issues and non-Gaussian noises in the dataset. In this work, we present an objective function based on the Kaniadakis κ-Gaussian distribution and the optimal transport (OT) theory to mitigate non-Gaussian noise effects and phase ambiguity concerns that cause cycle skipping. We construct the κ-objective function using the probabilistic maximum likelihood procedure and include it within a well-posed version of the original OT formulation, known as the Kantorovich-Rubinstein metric. We represent the data in the graph space to satisfy the probability axioms required by the Kantorovich-Rubinstein framework. We call our proposal the κ-Graph-Space Optimal Transport FWI (κ-GSOT-FWI). The results suggest that the κ-GSOT-FWI is an effective procedure to circumvent the effects of non-Gaussian noise and cycle-skipping problems. They also show that the Kaniadakis κ-statistics significantly improve the FWI objective function convergence, resulting in higher-resolution models than classical techniques, especially when κ=0.6.

Optimal Transport Approach to Sobolev Regularity of Solutions to the Weighted Least Gradient Problem

Optimal Transport Approach to Sobolev Regularity of Solutions to the Weighted Least Gradient Problem

A non-commutative entropic optimal transport approach to quantum composite systems at positive temperature

This paper establishes new connections between many-body quantum systems, One-body Reduced Density Matrices Functional Theory (1RDMFT) and Optimal Transport (OT), by interpreting the problem of computing the ground-state energy of a finite-dimensional composite quantum system at positive temperature as a non-commutative entropy regularized Optimal Transport problem. We develop a new approach to fully characterize the dual-primal solutions in such non-commutative setting. The mathematical formalism is particularly relevant in quantum chemistry: numerical realizations of the many-electron ground-state energy can be computed via a non-commutative version of Sinkhorn algorithm. Our approach allows to prove convergence and robustness of this algorithm, which, to our best knowledge, were unknown even in the two marginal case. Our methods are based on a priori estimates in the dual problem, which we believe to be of independent interest. Finally, the above results are extended in 1RDMFT setting, where bosonic or fermionic symmetry conditions are enforced on the problem.

Limitations of Variational Quantum Algorithms: A Quantum Optimal Transport Approach

The impressive progress in quantum hardware in the last years has raised the interest of the quantum computing community in harvesting the computational power of such devices. However, in the absence of error correction, these devices can only reliably implement very shallow circuits or comparatively deeper circuits at the expense of a nontrivial density of errors. In this work, we obtain extremely tight limitation bounds for standard NISQ proposals in both the noisy and noiseless regimes, with or without error-mitigation tools. The bounds limit the performance of both circuit model algorithms, such as QAOA, and also continuous-time algorithms, such as quantum annealing. In the noisy regime with local depolarizing noise $p$, we prove that at depths $L=\mathcal{O}(p^{-1})$ it is exponentially unlikely that the outcome of a noisy quantum circuit outperforms efficient classical algorithms for combinatorial optimization problems like Max-Cut. Although previous results already showed that classical algorithms outperform noisy quantum circuits at constant depth, these results only held for the expectation value of the output. Our results are based on newly developed quantum entropic and concentration inequalities, which constitute a homogeneous toolkit of theoretical methods from the quantum theory of optimal mass transport whose potential usefulness goes beyond the study of variational quantum algorithms.

Topological Speed Limit.

Any physical system evolves at a finite speed that is constrained not only by the energetic cost but also by the topological structure of the underlying dynamics. In this Letter, by considering such structural information, we derive a unified topological speed limit for the evolution of physical states using an optimal transport approach. We prove that the minimum time required for changing states is lower bounded by the discrete Wasserstein distance, which encodes the topological information of the system, and the time-averaged velocity. The bound obtained is tight and applicable to a wide range of dynamics, from deterministic to stochastic, and classical to quantum systems. In addition, the bound provides insight into the design principles of the optimal process that attains the maximum speed. We demonstrate the application of our results to chemical reaction networks and interacting many-body quantum systems.

Topics in robust statistical learning

Some recent contributions to robust inference are presented. Firstly, the classical problem of robust M-estimation of a location parameter is revisited using an optimal transport approach - with specifically designed Wasserstein-type distances - that reduces robustness to a continuity property. Secondly, a procedure of estimation of the distance function to a compact set is described, using union of balls. This methodology originates in the field of topological inference and offers as a byproduct a robust clustering method. Thirdly, a robust Lloyd-type algorithm for clustering is constructed, using a bootstrap variant of the median-of-means strategy. This algorithm comes with a robust initialization.

Gaussian Blue Noise

Among the various approaches for producing point distributions with blue noise spectrum, we argue for an optimization framework using Gaussian kernels. We show that with a wise selection of optimization parameters, this approach attains unprecedented quality, provably surpassing the current state of the art attained by the optimal transport (BNOT) approach. Further, we show that our algorithm scales smoothly and feasibly to high dimensions while maintaining the same quality, realizing unprecedented high-quality high-dimensional blue noise sets. Finally, we show an extension to adaptive sampling.

An Optimal Transport Approach to Deep Metric Learning (Student Abstract)

Capturing visual similarity among images is the core of many computer vision and pattern recognition tasks. This problem can be formulated in such a paradigm called metric learning. Most research in the area has been mainly focusing on improving the loss functions and similarity measures. However, due to the ignoring of geometric structure, existing methods often lead to sub-optimal results. Thus, several recent research methods took advantage of Wasserstein distance between batches of samples to characterize the spacial geometry. Although these approaches can achieve enhanced performance, the aggregation over batches definitely hinders Wasserstein distance's superior measure capability and leads to high computational complexity. To address this limitation, we propose a novel Deep Wasserstein Metric Learning framework, which employs Wasserstein distance to precisely capture the relationship among various images under ranking-based loss functions such as contrastive loss and triplet loss. Our method directly computes the distance between images, considering the geometry at a finer granularity than batch level. Furthermore, we introduce a new efficient algorithm using Sinkhorn approximation and Wasserstein measure coreset. The experimental results demonstrate the improvements of our framework over various baselines in different applications and benchmark datasets.

An Optimal Transport Approach for Selecting a Representative Subsample with Application in Efficient Kernel Density Estimation

Subsampling methods aim to select a subsample as a surrogate for the observed sample. Such methods have been used pervasively in large-scale data analytics, active learning, and privacy-preserving analysis in recent decades. Instead of model-based methods, in this article, we study model-free subsampling methods, which aim to identify a subsample, that is, not confined by model assumptions. Existing model-free subsampling methods are usually built upon clustering techniques or kernel tricks. Most of these methods suffer from either a large computational burden or a theoretical weakness. In particular, the theoretical weakness is that the empirical distribution of the selected subsample may not necessarily converge to the population distribution. Such computational and theoretical limitations hinder the broad applicability of model-free subsampling methods in practice. We propose a novel model-free subsampling method by using optimal transport techniques. Moreover, we develop an efficient subsampling algorithm, that is, adaptive to the unknown probability density function. Theoretically, we show the selected subsample can be used for efficient density estimation by deriving the convergence rate for the proposed subsample kernel density estimator. We also provide the optimal bandwidth for the proposed estimator. Numerical studies on synthetic and real-world datasets demonstrate the performance of the proposed method is superior.

An Optimal Transport Approach to Personalized Federated Learning

Federated learning is a distributed machine learning paradigm, which aims to train a model using the local data of many distributed clients. A key challenge in federated learning is that the data samples across the clients may not be identically distributed. To address this challenge, personalized federated learning with the goal of tailoring the learned model to the data distribution of every individual client has been proposed. In this paper, we focus on this problem and propose a novel personalized Federated Learning scheme based on Optimal Transport (FedOT) as a learning algorithm that learns the optimal transport maps for transferring data points to a common distribution as well as the prediction model under the applied transport map. To formulate the FedOT problem, we extend the standard optimal transport task between two probability distributions to multi-marginal optimal transport problems with the goal of transporting samples from multiple distributions to a common probability domain. We then leverage the results on multi-marginal optimal transport problems to formulate FedOT as a min-max optimization problem and analyze its generalization and optimization properties. We discuss the results of several numerical experiments to evaluate the performance of FedOT under heterogeneous data distributions in federated learning problems.

Quantitative observability for the Schrödinger and Heisenberg equations: An optimal transport approach

We establish a quantitative observation inequality for the Schrödinger and the Heisenberg equations on [Formula: see text], uniform in the Planck constant [Formula: see text]. The proof is based on the pseudometric introduced in F. Golse and T. Paul, The Schrödinger equation in the mean-field and semiclassical regime, Arch. Ration. Mech. Anal. 223 (2017) 57–94. This inequality involves only effective constants which are computed explicitly in their dependence in [Formula: see text] and all parameters involved.

Computational optimal transport for molecular spectra: The semi-discrete case.

Comparing a discrete molecular spectrum to a continuous molecular spectrum in a quantitative manner is a challenging problem, for example, when attempting to fit a theoretical stick spectrum to a continuous spectrum. In this paper, the use of computational optimal transport is investigated for such a problem. In the optimal transport literature, the comparison of a discrete and a continuous spectrum is referred to as semi-discrete optimal transport and is a situation where a metric such as least-squares may be difficult to define except under special conditions. The merits of an optimal transport approach for this problem are investigated using the transport distance defined for the semi-discrete case. A tutorial on semi-discrete optimal transport for molecular spectra is included in this paper, and several well-chosen synthetic spectra are investigated to demonstrate the utility of computational optimal transport for the semi-discrete case. Among several types of investigations, we include calculations showing how the frequency resolution of the continuous spectrum affects the transport distance between a discrete and a continuous spectrum. We also use the transport distance to measure the distance between a continuous experimental electronic absorption spectrum of SO2 and a theoretical stick spectrum for the same system. The comparison of the theoretical and experimental SO2 spectra also allows us to suggest a theoretical value for the band origin that is closer to the observed band origin than previous theoretical values.

Measure estimation on manifolds: an optimal transport approach

Assume that we observe i.i.d. points lying close to some unknown d-dimensional \({\mathcal {C}}^k\) submanifold M in a possibly high-dimensional space. We study the problem of reconstructing the probability distribution generating the sample. After remarking that this problem is degenerate for a large class of standard losses (\(L_p\), Hellinger, total variation, etc.), we focus on the Wasserstein loss, for which we build an estimator, based on kernel density estimation, whose rate of convergence depends on d and the regularity \(s\le k-1\) of the underlying density, but not on the ambient dimension. In particular, we show that the estimator is minimax and matches previous rates in the literature in the case where the manifold M is a d-dimensional cube. The related problem of the estimation of the volume measure of M for the Wasserstein loss is also considered, for which a minimax estimator is exhibited.

DOTA: Deep Learning Optimal Transport Approach to Advance Drug Repositioning for Alzheimer's Disease.

Alzheimer’s disease (AD) is the leading cause of age-related dementia, affecting over 5 million people in the United States and incurring a substantial global healthcare cost. Unfortunately, current treatments are only palliative and do not cure AD. There is an urgent need to develop novel anti-AD therapies; however, drug discovery is a time-consuming, expensive, and high-risk process. Drug repositioning, on the other hand, is an attractive approach to identify drugs for AD treatment. Thus, we developed a novel deep learning method called DOTA (Drug repositioning approach using Optimal Transport for Alzheimer’s disease) to repurpose effective FDA-approved drugs for AD. Specifically, DOTA consists of two major autoencoders: (1) a multi-modal autoencoder to integrate heterogeneous drug information and (2) a Wasserstein variational autoencoder to identify effective AD drugs. Using our approach, we predict that antipsychotic drugs with circadian effects, such as quetiapine, aripiprazole, risperidone, suvorexant, brexpiprazole, olanzapine, and trazadone, will have efficacious effects in AD patients. These drugs target important brain receptors involved in memory, learning, and cognition, including serotonin 5-HT2A, dopamine D2, and orexin receptors. In summary, DOTA repositions promising drugs that target important biological pathways and are predicted to improve patient cognition, circadian rhythms, and AD pathogenesis.