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The optimal transport barycenter (a.k.a. Wasserstein barycenter) is a fundamental notion of averaging that extends from the Euclidean space to the Wasserstein space of probability distributions. Computation of the unregularized barycenter for discretized probability distributions on point clouds is a challenging task when the domain dimension d>1. Most practical algorithms for approximating the barycenter problem are based on entropic regularization. In this paper, we introduce a nearly linear time O(mlogm) and linear space complexity O(m) primal-dual algorithm, the Wasserstein-Descent ℍ˙1-Ascent (WDHA) algorithm, for computing the exact barycenter when the input probability density functions are discretized on an m-point grid. The key success of the WDHA algorithm hinges on alternating between two different yet closely related Wasserstein and Sobolev optimization geometries for the primal barycenter and dual Kantorovich potential subproblems. Under reasonable assumptions, we establish the convergence rate and iteration complexity of WDHA to its stationary point when the step size is appropriately chosen. Superior computational efficacy, scalability, and accuracy over the existing Sinkhorn-type algorithms are demonstrated on high-resolution (e.g., 1024×1024 images) 2D synthetic and real data. 

Distributional data have become increasingly prominent in modern signal processing, highlighting the necessity of computing optimal transport (OT) maps across multiple probability distributions. Nevertheless, recent studies on neural OT methods predominantly focused on the efficient computation of a single map between two distributions. To address this challenge, we introduce a novel approach to learning transport maps for new empirical distributions. Specifically, we employ the transformer architecture to produce embeddings from distributional data of varying length; these embeddings are then fed into a hypernetwork to generate neural OT maps. Various numerical experiments were conducted to validate the embeddings and the generated OT maps. 

Optimal transport (OT) offers a versatile framework to compare complex data distributions in a geometrically meaningful way. Traditional methods for computing the Wasserstein distance and geodesic between probability measures require mesh-specific domain discretization and suffer from the curse-of-dimensionality. We present GeONet, a mesh-invariant deep neural operator network that learns the non-linear mapping from the input pair of initial and terminal distributions to the Wasserstein geodesic connecting the two endpoint distributions. In the offline training stage, GeONet learns the saddle point optimality conditions for the dynamic formulation of the OT problem in the primal and dual spaces that are characterized by a coupled PDE system. The subsequent inference stage is instantaneous and can be deployed for real-time predictions in the online learning setting. We demonstrate that GeONet achieves comparable testing accuracy to the standard OT solvers on simulation examples and the MNIST dataset with considerably reduced inference-stage computational cost by orders of magnitude. 

Motivated by approximation Bayesian computation using mean-field variational approximation and the computation of equilibrium in multi-species systems with cross-interaction, this paper investigates the composite geodesically convex optimization problem over multiple distributions. The objective functional under consideration is composed of a convex potential energy on a product of Wasserstein spaces and a sum of convex self-interaction and internal energies associated with each distribution. To efficiently solve this problem, we introduce the Wasserstein Proximal Coordinate Gradient (WPCG) algorithms with parallel, sequential, and random update schemes. Under a quadratic growth (QG) condition that is weaker than the usual strong convexity requirement on the objective functional, we show that WPCG converges exponentially fast to the unique global optimum. In the absence of the QG condition, WPCG is still demonstrated to converge to the global optimal solution, albeit at a slower polynomial rate. Numerical results for both motivating examples are consistent with our theoretical findings. 

Motivated by approximation Bayesian computation using mean-field variational approximation and the computation of equilibrium in multi-species systems with cross-interaction, this paper investigates the composite geodesically convex optimization problem over multiple distributions. The objective functional under consideration is composed of a convex potential energy on a product of Wasserstein spaces and a sum of convex self-interaction and internal energies associated with each distribution. To efficiently solve this problem, we introduce the Wasserstein Proximal Coordinate Gradient (WPCG) algorithms with parallel, sequential, and random update schemes. Under a quadratic growth (QG) condition that is weaker than the usual strong convexity requirement on the objective functional, we show that WPCG converges exponentially fast to the unique global optimum. In the absence of the QG condition, WPCG is still demonstrated to converge to the global optimal solution, albeit at a slower polynomial rate. Numerical results for both motivating examples are consistent with our theoretical findings. 

Extensive research in Medical Imaging aims to uncover critical diagnostic features in patients, with AI-driven medical diagnosis relying on sophisticated machine learning and deep learning models to analyze, detect, and identify diseases from medical images. Despite the remarkable accuracy of these models under normal conditions, they grapple with trustworthiness issues, where their output could be manipulated by adversaries who introduce strategic perturbations to the input images. Furthermore, the scarcity of publicly available medical images, constituting a bottleneck for reliable training, has led contemporary algorithms to depend on pretrained models grounded on a large set of natural images—a practice referred to as transfer learning. However, a significant domain discrepancy exists between natural and medical images, which causes AI models resulting from transfer learning to exhibit heightened vulnerability to adversarial attacks. This paper proposes a domain assimilation approach that introduces texture and color adaptation into transfer learning, followed by a texture preservation component to suppress undesired distortion. We systematically analyze the performance of transfer learning in the face of various adversarial attacks under different data modalities, with the overarching goal of fortifying the model’s robustness and security in medical imaging tasks. The results demonstrate high effectiveness in reducing attack efficacy, contributing toward more trustworthy transfer learning in biomedical applications. 

Extensive research in Medical Imaging aims to uncover critical diagnostic features in patients, with AI-driven medical diagnosis relying on sophisticated machine learning and deep learning models to analyze, detect, and identify diseases from medical images. Despite the remarkable accuracy of these models under normal conditions, they grapple with trustworthiness issues, where their output could be manipulated by adversaries who introduce strategic perturbations to the input images. Furthermore, the scarcity of publicly available medical images, constituting a bottleneck for reliable training, has led contemporary algorithms to depend on pretrained models grounded on a large set of natural images—a practice referred to as transfer learning. However, a significant domain discrepancy exists between natural and medical images, which causes AI models resulting from transfer learning to exhibit heightened vulnerability to adversarial attacks. This paper proposes a domain assimilation approach that introduces texture and color adaptation into transfer learning, followed by a texture preservation component to suppress undesired distortion. We systematically analyze the performance of transfer learning in the face of various adversarial attacks under different data modalities, with the overarching goal of fortifying the model’s robustness and security in medical imaging tasks. The results demonstrate high effectiveness in reducing attack efficacy, contributing toward more trustworthy transfer learning in biomedical applications.