A geometric view of optimal transportation and generative model

Lei, Na; Su, Kehua; Li, Cui; Yau, Shing–Tung; Gu, Xianfeng

doi:10.1016/j.cagd.2018.10.005

Cited by 95 publications

(46 citation statements)

References 45 publications

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“…Gu [41] proposed a discrete spherical optimal transportation mapping based on a purely geometric method and defined the measures as areas to achieve an area-preserving mapping from the topological sphere to the unit sphere [42,43]. The mapping is global and has been applied in biomedicine [44,45], face recognition [46], generative adversarial networks [47], and other fields.…”

Section: Theory and Methodsmentioning

confidence: 99%

An Overall Uniformity Optimization Method of the Spherical Icosahedral Grid Based on the Optimal Transformation Theory

et al. 2021

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The improvement of overall uniformity and smoothness of spherical icosahedral grids, the basic framework of atmospheric models, is a key to reducing simulation errors. However, most of the existing grid optimization methods have optimized grid from different aspects and not improved overall uniformity and smoothness of grid at the same time, directly affecting the accuracy and stability of numerical simulation. Although a well-defined grid with more than 12 points cannot be constructed on a sphere, the area uniformity and the interval uniformity of the spherical grid can be traded off to enhance extremely the overall grid uniformity and smoothness. To solve this problem, an overall uniformity and smoothness optimization method of the spherical icosahedral grid is proposed based on the optimal transformation theory. The spherical cell decomposition method has been introduced to iteratively update the grid to minimize the spherical transportation cost, achieving an overall optimization of the spherical icosahedral grid. Experiments on the four optimized grids (the spring dynamics optimized grid, the Heikes and Randall optimized grid, the spherical centroidal Voronoi tessellations optimized grid and XU optimized grid) demonstrate that the grid area uniformity of our method has been raised by 22.60% of SPRG grid, −1.30% of HR grid, 38.30% of SCVT grid and 38.20% of XU grid, and the grid interval uniformity has been improved by 2.50% of SPRG grid, 2.80% of HR grid, 11.10% of SCVT grid and 11.00% of XU grid. Although the grid uniformity of the proposed method is similar with the HR grid, the smoothness of grid deformation has been enhanced by 79.32% of grid area and 24.07% of grid length. To some extent, the proposed method may be viewed as a novel optimization approach of the spherical icosahedral grid which can improve grid overall uniformity and smoothness of grid deformation.

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Section: Theory and Methodsmentioning

confidence: 99%

An Overall Uniformity Optimization Method of the Spherical Icosahedral Grid Based on the Optimal Transformation Theory

et al. 2021

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“…In theory, we need to first study whether there exists an optimal map for the manifold setting. For optimal mass transport on realvalued data, (Evans 1997;Villani 2008;Lei et al 2017) proved the existence of the optimal map, which has a convex potential (i.e. D(x) = ∇φ(x) with φ(x) being convex) and is shown to be the only map with the convex potential.…”

Section: Problem Formulationmentioning

confidence: 99%

Manifold-Valued Image Generation with Wasserstein Generative Adversarial Nets

Huang

Gool

2019

AAAI

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Generative modeling over natural images is one of the most fundamental machine learning problems. However, few modern generative models, including Wasserstein Generative Adversarial Nets (WGANs), are studied on manifold-valued images that are frequently encountered in real-world applications. To fill the gap, this paper first formulates the problem of generating manifold-valued images and exploits three typical instances: hue-saturation-value (HSV) color image generation, chromaticity-brightness (CB) color image generation, and diffusion-tensor (DT) image generation. For the proposed generative modeling problem, we then introduce a theorem of optimal transport to derive a new Wasserstein distance of data distributions on complete manifolds, enabling us to achieve a tractable objective under the WGAN framework. In addition, we recommend three benchmark datasets that are CIFAR-10 HSV/CB color images, ImageNet HSV/CB color images, UCL DT image datasets. On the three datasets, we experimentally demonstrate the proposed manifold-aware WGAN model can generate more plausible manifold-valued images than its competitors.

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“…Gu et al utilize Optimal Transportation and Monge-Ampere equation to theoretically interpret deep learning and GAN [51]- [53]. They show the intrinsic relations between optimal transportation and convex geometry, the generator calculates the transportation map while the discriminator computes the Kantorovich potential [52].…”

Section: B Generative Adversarial Networkmentioning

confidence: 99%

Improving Human Pose Estimation With Self-Attention Generative Adversarial Networks

et al. 2019

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Human pose estimation in images is challenging and important for many computer vision applications. Large improvements in human pose estimation have been achieved with the development of convolutional neural networks. Even though, when encountered some difficult cases even the state-ofthe-art models may fail to predict all the body joints correctly. Some recent works try to refine the pose estimator. GAN (Generative Adversarial Networks) has been proved to be efficient to improve human pose estimation. However, GAN can only learn local body joints structural constrains. In this paper, we propose to apply Self-Attention GAN to further improve the performance of human pose estimation. With attention mechanism in the framework of GAN, we can learn long-range body joints dependencies, therefore enforce the entire body joints structural constrains to make all the body joints to be consistent. Our method outperforms other state-of-the-art methods on two standard benchmark datasets MPII and LSP for human pose estimation. Our code is available at: https://github.com/idotc/Hg-SAGAN. INDEX TERMS Human pose estimation, convolutional neural networks, stacked hourglass networks, self-attention GAN.

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A geometric view of optimal transportation and generative model

Cited by 95 publications

References 45 publications

An Overall Uniformity Optimization Method of the Spherical Icosahedral Grid Based on the Optimal Transformation Theory

An Overall Uniformity Optimization Method of the Spherical Icosahedral Grid Based on the Optimal Transformation Theory

Manifold-Valued Image Generation with Wasserstein Generative Adversarial Nets

Improving Human Pose Estimation With Self-Attention Generative Adversarial Networks

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