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1. A geometric variational framework for computing spherical optimal transportation maps II

   https://doi.org/10.4310/MCGD.2022.v2.n2.a2  

   Zhao, Zhou; Lei, Na; Cui, Li; Su, Kehua; Xu, Xiaoyin; Luo, Feng; Gu, Xianfeng; Yau, Shing-Tung (January 2022, Mathematics, Computation and Geometry of Data)

   Optimal transportation maps play fundamental roles in many engineering and medical fields. The computation of optimal transportation maps can be reduced to solve highly non-linear Monge-Ampere equations. This work summarizes the geometric variational frameworks for spherical optimal transportation maps, which offers solutions to the Minkowski problem in convex differential geometry, reflector design and refractor design problems in optics. The method is rigorous, robust and efficient. The algorithm can directly generalized to higher dimensions. 

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2. FFT-OT: A Fast Algorithm for Optimal Transportation

   Lei, Na; Gu, Xianfeng (July 2021, Proceedings of International Conference on Computer Vision (ICCV))

   null (Ed.)

   An optimal transportation map finds the most economical way to transport one probability measure to the other. It has been applied in a broad range of applications in vision, deep learning and medical images. By Brenier theory, computing the optimal transport map is equivalent to solving a Monge-Amp\`ere equation. Due to the highly non-linear nature, the computation of optimal transportation maps in large scale is very challenging. This work proposes a simple but powerful method, the FFT-OT algorithm, to tackle this difficulty based on three key ideas. First, solving Monge-Amp\`ere equation is converted to a fixed point problem; Second, the obliqueness property of optimal transportation maps are reformulated as Neumann boundary conditions on rectangular domains; Third, FFT is applied in each iteration to solve a Poisson equation in order to improve the efficiency. Experiments on surfaces captured from 3D scanning and reconstructed from medical imaging are conducted, and compared with other existing methods. Our experimental results show that the proposed FFT-OT algorithm is simple, general and scalable with high efficiency and accuracy. 

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3. Cortical Morphometry Analysis based on Worst Transportation Theory

   Zhang, Min; An, Dongsheng; Lei, Na; Wu, Jianfeng; Zhao, Tong; Xu, Xiaoyin; Wang, Yalin; Gu, Xianfeng (October 2021, IPMI 2021: Information Processing in Medical Imaging)

   null (Ed.)

   Biomarkers play an important role in early detection and intervention in Alzheimer’s disease (AD). However, obtaining effective biomarkers for AD is still a big challenge. In this work, we propose to use the worst transportation cost as a univariate biomarker to index cortical morphometry for tracking AD progression. The worst transportation (WT) aims to find the least economical way to transport one measure to the other, which contrasts to the optimal transportation (OT) that finds the most economical way between measures. To compute the WT cost, we generalize the Brenier theorem for the OT map to the WT map, and show that the WT map is the gradient of a concave function satisfying the Monge-Ampere equation. We also develop an efficient algorithm to compute the WT map based on computational geometry. We apply the algorithm to analyze cortical shape difference between dementia due to AD and normal aging individuals. The experimental results reveal the effectiveness of our proposed method which yields better statistical performance than other competiting methods including the OT. 

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4. AE-OT: A new generative Model Based on Extended Semi-discrete Optimal Transport

   An, Dongsheng; Guo, Yang; Lei, Na; Luo, Zhongxuan; Yau, Shing-Tung; Gu, Xianfeng. (January 2019, ICLR 2020)

   Generative adversarial networks (GANs) have attracted huge attention due to its capability to generate visual realistic images. However, most of the existing models suffer from the mode collapse or mode mixture problems. In this work, we give a theoretic explanation of the both problems by Figalli’s regularity theory of optimal transportation maps. Basically, the generator compute the transportation maps between the white noise distributions and the data distributions, which are in general discontinuous. However, DNNs can only represent continuous maps. This intrinsic conflict induces mode collapse and mode mixture. In order to tackle the both problems, we explicitly separate the manifold embedding and the optimal transportation; the first part is carried out using an autoencoder to map the images onto the latent space; the second part is accomplished using a GPU-based convex optimization to find the discontinuous transportation maps. Composing the extended OT map and the decoder, we can finally generate new images from the white noise. This AE-OT model avoids representing discontinuous maps by DNNs, therefore effectively prevents mode collapse and mode mixture. 

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5. Mobility Equity and Economic Sustainability Using Game Theory

   Chremos, I.V.; Malikopoulos, A.A. (May 2023, 2023 American Control Conference)

   In this paper, we consider a multi-modal mobility system of travelers each with an individual travel budget, and propose a game-theoretic framework to assign each traveler to a “mobility service” (each one representing a different mode of transportation). We are interested in equity and sustainability, thus we maximize the worst-case revenue of the mobility system while ensuring “mobility equity,” which we define it in terms of accessibility. In the proposed framework, we ensure that all travelers are truthful and voluntarily participate under informational asymmetry, and the solution respects the individual budget of each traveler. Each traveler may seek to travel using multiple services (e.g., car, bus, train, bike). The services are capacitated and can serve up to a fixed number of travelers at any instant of time. Thus, our problem falls under the category of many-to-one assignment problems, where the goal is to find the conditions that guarantee the stability of assignments. We formulate a linear program of maximizing worst-case revenue under the constraints of mobility equity, and we fully characterize the optimal solution. 

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