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Many complex networks in the real world have community structures – groups of well-connected nodes with important functional roles. It has been well recognized that the identification of communities bears numerous practical applications. While existing approaches mainly apply statistical or graph theoretical/combinatorial methods for community detection, in this paper, we present a novel geometric approach which  enables us to borrow powerful classical geometric methods and properties. By considering networks as geometric objects and communities in a network as a geometric decomposition, we apply curvature and discrete Ricci flow, which have been used to decompose smooth manifolds with astonishing successes in mathematics, to break down communities in networks. We  tested our method on networks with ground-truth community structures, and experimentally confirmed the effectiveness of this geometric approach.

 

Surface registration plays a fundamental role in shape analysis and geometric processing. Generally, there are three criteria in evaluating a surface mapping result: diffeomorphism, small distortion, and feature alignment. To fulfill these requirements, this work proposes a novel model of the space of point landmark constrained diffeomorphisms. Based on Teichm¨uller theory, this mapping space is generated by the Beltrami coefficients, which are infinitesimally Teichm¨uller equivalent to 0. These Beltrami coefficients are the solutions to a linear equation group. By using this theoretic model, optimal registrations can be achieved by iterative optimization with linear constraints in the diffeomorphism space, such as harmonic maps and Teichm¨uller maps, which minimize different types of distortion. The theoretical model is rigorous and has practical value. Our experimental results demonstrate the efficiency and efficacy of the proposed method. 

Though generative adversarial networks (GANs) are prominent models to generate realistic and crisp images, they are unstable to train and suffer from the mode collapse problem. The problems of GANs come from approximating the intrinsic discontinuous distribution transform map with continuous DNNs. The recently proposed AE-OT model addresses the discontinuity problem by explicitly computing the discontinuous optimal transform map in the latent space of the autoencoder. Though have no mode collapse, the generated images by AE-OT are blurry. In this paper, we propose the AE-OT-GAN model to utilize the advantages of the both models: generate high quality images and at the same time overcome the mode collapse problems. Specifically, we firstly embed the low dimensional image manifold into the latent space by autoencoder (AE). Then the extended semi-discrete optimal transport (SDOT) map is used to generate new latent codes. Finally, our GAN model is trained to generate high quality images from the latent distribution induced by the extended SDOT map. The distribution transform map from this dataset related latent distribution to the data distribution will be continuous, and thus can be well approximated by the continuous DNNs. Additionally, the paired data between the latent codes and the real images gives us further restriction about the generator and stabilizes the training process. Experiments on simple MNIST dataset and complex datasets like CIFAR10 and CelebA show the advantages of the proposed method.