On the Imbedding Problem for Three-state Time Homogeneous Markov Chains with Coinciding Negative Eigenvalues

For an indecomposable $3\times 3$ stochastic matrix (i.e., 1-step transition probability matrix) with coinciding negative eigenvalues, a new necessary and sufficient condition of the imbedding problem for time homogeneous Markov chains is shown by means of an alternate parameterization of the transition rate matrix (i.e., intensity matrix, infinitesimal generator), which avoids calculating matrix logarithm or matrix square root. In addition, an implicit description of the imbedding problem for the $3\times 3$ stochastic matrix in Johansen [J. Lond. Math. Soc., 8, 345-351. (1974)] is pointed out.Comment: 17 page

Weighted projective lines of tubular type and equivariantization

We prove that the categories of coherent sheaves over weighted projective lines of tubular type are explicitly related to each other via the equivariantization with respect to certain cyclic group actions

Pr\"ufer sheaves and generic sheaves over the weighted projective lines and elliptic curves

In the present paper, we introduce the concepts of Pr\"{u}fer sheaves and adic sheaves over a weighted projective line of genus one or an elliptic curve, show that Pr\"{u}fer sheaves and adic sheaves can classify the category of coherent sheaves. Moreover, we describe the relationship between Pr\"{u}fer sheaves and generic sheaves, and provide two methods to construct generic sheaves by using coherent sheaves and Pr\"{u}fer sheaves.Comment: 27 page

Monadicity theorem and weighted projective lines of tubular type

We formulate a version of Beck's monadicity theorem for abelian categories, which is applied to the equivariantization of abelian categories with respect to a finite group action. We prove that the equivariantization is compatible with the construction of quotient abelian categories by Serre subcategories. We prove that the equivariantization of the graded module category over a graded ring is equivalent to the graded module category over the same ring but with a different grading. We deduce from these results two equivalences between the category of (equivariant) coherent sheaves on a weighted projective line of tubular type and that on an elliptic curve, where the acting groups are cyclic and the two equivalences are somehow adjoint to each other.Comment: 3 table

GeoCapsNet: Aerial to Ground view Image Geo-localization using Capsule Network

The task of cross-view image geo-localization aims to determine the geo-location (GPS coordinates) of a query ground-view image by matching it with the GPS-tagged aerial (satellite) images in a reference dataset. Due to the dramatic changes of viewpoint, matching the cross-view images is challenging. In this paper, we propose the GeoCapsNet based on the capsule network for ground-to-aerial image geo-localization. The network first extracts features from both ground-view and aerial images via standard convolution layers and the capsule layers further encode the features to model the spatial feature hierarchies and enhance the representation power. Moreover, we introduce a simple and effective weighted soft-margin triplet loss with online batch hard sample mining, which can greatly improve image retrieval accuracy. Experimental results show that our GeoCapsNet significantly outperforms the state-of-the-art approaches on two benchmark datasets

Tilting objects in the stable category of vector bundles on the weighted projective line of type (2,2,2,2;\lambda)

We construct a tilting object for the stable category of vector bundles on a weighted projective line X of type (2,2,2,2;\lambda), consisting of five rank two bundles and one rank three bundle, whose endomorphism algebra is a canonical algebra associated with X of type (2,2,2,2)

Towards Open-Set Identity Preserving Face Synthesis

We propose a framework based on Generative Adversarial Networks to disentangle the identity and attributes of faces, such that we can conveniently recombine different identities and attributes for identity preserving face synthesis in open domains. Previous identity preserving face synthesis processes are largely confined to synthesizing faces with known identities that are already in the training dataset. To synthesize a face with identity outside the training dataset, our framework requires one input image of that subject to produce an identity vector, and any other input face image to extract an attribute vector capturing, e.g., pose, emotion, illumination, and even the background. We then recombine the identity vector and the attribute vector to synthesize a new face of the subject with the extracted attribute. Our proposed framework does not need to annotate the attributes of faces in any way. It is trained with an asymmetric loss function to better preserve the identity and stabilize the training process. It can also effectively leverage large amounts of unlabeled training face images to further improve the fidelity of the synthesized faces for subjects that are not presented in the labeled training face dataset. Our experiments demonstrate the efficacy of the proposed framework. We also present its usage in a much broader set of applications including face frontalization, face attribute morphing, and face adversarial example detection

Ranking the Importance of Nodes of Complex Networks by the Equivalence Classes Approach

Identifying the importance of nodes of complex networks is of interest to the research of Social Networks, Biological Networks etc.. Current researchers have proposed several measures or algorithms, such as betweenness, PageRank and HITS etc., to identify the node importance. However, these measures are based on different aspects of properties of nodes, and often conflict with the others. A reasonable, fair standard is needed for evaluating and comparing these algorithms. This paper develops a framework as the standard for ranking the importance of nodes. Four intuitive rules are suggested to measure the node importance, and the equivalence classes approach is employed to resolve the conflicts and aggregate the results of the rules. To quantitatively compare the algorithms, the performance indicators are also proposed based on a similarity measure. Three widely used real-world networks are used as the test-beds. The experimental results illustrate the feasibility of this framework and show that both algorithms, PageRank and HITS, perform well with bias when dealing with the tested networks. Furthermore, this paper uses the proposed approach to analyze the structure of the Internet, and draws out the kernel of the Internet with dense links

CVAE-GAN: Fine-Grained Image Generation through Asymmetric Training

We present variational generative adversarial networks, a general learning framework that combines a variational auto-encoder with a generative adversarial network, for synthesizing images in fine-grained categories, such as faces of a specific person or objects in a category. Our approach models an image as a composition of label and latent attributes in a probabilistic model. By varying the fine-grained category label fed into the resulting generative model, we can generate images in a specific category with randomly drawn values on a latent attribute vector. Our approach has two novel aspects. First, we adopt a cross entropy loss for the discriminative and classifier network, but a mean discrepancy objective for the generative network. This kind of asymmetric loss function makes the GAN training more stable. Second, we adopt an encoder network to learn the relationship between the latent space and the real image space, and use pairwise feature matching to keep the structure of generated images. We experiment with natural images of faces, flowers, and birds, and demonstrate that the proposed models are capable of generating realistic and diverse samples with fine-grained category labels. We further show that our models can be applied to other tasks, such as image inpainting, super-resolution, and data augmentation for training better face recognition models.Comment: to appear in ICCV 201

Revisiting Distributed Synchronous SGD

Distributed training of deep learning models on large-scale training data is typically conducted with asynchronous stochastic optimization to maximize the rate of updates, at the cost of additional noise introduced from asynchrony. In contrast, the synchronous approach is often thought to be impractical due to idle time wasted on waiting for straggling workers. We revisit these conventional beliefs in this paper, and examine the weaknesses of both approaches. We demonstrate that a third approach, synchronous optimization with backup workers, can avoid asynchronous noise while mitigating for the worst stragglers. Our approach is empirically validated and shown to converge faster and to better test accuracies.Comment: 10 page