Scaling limit of the local time of the (1,L)−(1,L)-random walk

It is well known (Donsker's Invariance Principle) that the random walk converges to Brownian motion by scaling. In this paper, we will prove that the scaled local time of the $(1,L)-$random walk converges to that of the Brownian motion. The results was proved by Rogers (1984) in the case $L=1$. Our proof is based on the intrinsic multiple branching structure within the $(1,L)-$random walk revealed by Hong and Wang (2013)

Exact Asymptotic Behavior of Singular Positive Solutions of Fractional Semi-Linear Elliptic Equations

In this paper, we prove the exact asymptotic behavior of singular positive solutions of fractional semi-linear equations $$(-\Delta)^\sigma u = u^p~~~~~~~~in ~~ B_1\backslash \{0\}$$ with an isolated singularity, where $\sigma \in (0, 1)$ and $\frac{n}{n-2\sigma} < p < \frac{n+2\sigma}{n-2\sigma}$.Comment: 11 page

Qualitative analysis for an elliptic system in the punctured space

In this paper, we investigate the qualitative properties of positive solutions for the following two-coupled elliptic system in the punctured space: $$\begin{cases} -\Delta u =\mu_1 u^{2q+1} + \beta u^q v^{q+1} \\ -\Delta v =\mu_2 v^{2q+1} + \beta v^q u^{q+1} \end{cases} \textmd{in} ~\mathbb{R}^n \backslash \{0\},$$ where $\mu_1, \mu_2$ and $\beta$ are all positive constants, $n\geq 3$. We establish a monotonicity formula that completely characterizes the singularity of positive solutions. We prove a sharp global estimate for both components of positive solutions. We also prove the nonexistence of positive semi-singular solutions, which means that one component is bounded near the singularity and the other component is unbounded near the singularity.Comment: 27 page

Scaling limit of the local time of the Sinai's random walk

We prove that the local times of a sequence of Sinai's random walks convergence to those of Brox's diffusion by proper scaling, which is accord with the result of Seignourel (2000). Our proof is based on the convergence of the branching processes in random environment by Kurtz (1979)

Recurrent Regression for Face Recognition

To address the sequential changes of images including poses, in this paper we propose a recurrent regression neural network(RRNN) framework to unify two classic tasks of cross-pose face recognition on still images and video-based face recognition. To imitate the changes of images, we explicitly construct the potential dependencies of sequential images so as to regularize the final learning model. By performing progressive transforms for sequentially adjacent images, RRNN can adaptively memorize and forget the information that benefits for the final classification. For face recognition of still images, given any one image with any one pose, we recurrently predict the images with its sequential poses to expect to capture some useful information of others poses. For video-based face recognition, the recurrent regression takes one entire sequence rather than one image as its input. We verify RRNN in static face dataset MultiPIE and face video dataset YouTube Celebrities(YTC). The comprehensive experimental results demonstrate the effectiveness of the proposed RRNN method

CFSNet: Toward a Controllable Feature Space for Image Restoration

Deep learning methods have witnessed the great progress in image restoration with specific metrics (e.g., PSNR, SSIM). However, the perceptual quality of the restored image is relatively subjective, and it is necessary for users to control the reconstruction result according to personal preferences or image characteristics, which cannot be done using existing deterministic networks. This motivates us to exquisitely design a unified interactive framework for general image restoration tasks. Under this framework, users can control continuous transition of different objectives, e.g., the perception-distortion trade-off of image super-resolution, the trade-off between noise reduction and detail preservation. We achieve this goal by controlling the latent features of the designed network. To be specific, our proposed framework, named Controllable Feature Space Network (CFSNet), is entangled by two branches based on different objectives. Our framework can adaptively learn the coupling coefficients of different layers and channels, which provides finer control of the restored image quality. Experiments on several typical image restoration tasks fully validate the effective benefits of the proposed method. Code is available at https://github.com/qibao77/CFSNet.Comment: Accepted by ICCV 201

Tensor graph convolutional neural network

In this paper, we propose a novel tensor graph convolutional neural network (TGCNN) to conduct convolution on factorizable graphs, for which here two types of problems are focused, one is sequential dynamic graphs and the other is cross-attribute graphs. Especially, we propose a graph preserving layer to memorize salient nodes of those factorized subgraphs, i.e. cross graph convolution and graph pooling. For cross graph convolution, a parameterized Kronecker sum operation is proposed to generate a conjunctive adjacency matrix characterizing the relationship between every pair of nodes across two subgraphs. Taking this operation, then general graph convolution may be efficiently performed followed by the composition of small matrices, which thus reduces high memory and computational burden. Encapsuling sequence graphs into a recursive learning, the dynamics of graphs can be efficiently encoded as well as the spatial layout of graphs. To validate the proposed TGCNN, experiments are conducted on skeleton action datasets as well as matrix completion dataset. The experiment results demonstrate that our method can achieve more competitive performance with the state-of-the-art methods

On Isolated Singularities of Fractional Semi-Linear Elliptic Equations

In this paper, we study the local behavior of nonnegative solutions of fractional semi-linear equations $(-\Delta)^\sigma u = u^p$ with an isolated singularity, where \sg \in (0, 1) and \frac{n}{n-2\sg} < p < \frac{n+2\sg}{n-2\sg}. We first use blow up method and a Liouville type theorem to derive an upper bound. Then we establish a monotonicity formula and a sufficient condition for removable singularity to give a classification of the isolated singularities. When \sg=1, this classification result has been proved by Gidas and Spruck (Comm. Pure Appl. Math. 34: 525-598, 1981).Comment: 19 page

Scaling limit theorems for the κ\kappa-transient random walk in random and non-random environment

Kesten et al.( 1975) proved the stable law for the transient RWRE (here we refer it as the $\kappa$-transient RWRE). After that, some similar interesting properties have also been revealed for its continuous counterpart, the diffusion proces in a Brownian environment with drift $\kappa$. In the present paper we will investigate the connections between these two kind of models, i.e., we will construct a sequence of the $\kappa$-transient RWREs and prove it convergence to the diffusion proces in a Brownian environment with drift $\kappa$ by proper scaling. To this end, we need a counterpart convergence for the $\kappa$-transient random walk in non-random environment, which is interesting itself

Spatial-Temporal Recurrent Neural Network for Emotion Recognition

Emotion analysis is a crucial problem to endow artifact machines with real intelligence in many large potential applications. As external appearances of human emotions, electroencephalogram (EEG) signals and video face signals are widely used to track and analyze human's affective information. According to their common characteristics of spatial-temporal volumes, in this paper we propose a novel deep learning framework named spatial-temporal recurrent neural network (STRNN) to unify the learning of two different signal sources into a spatial-temporal dependency model. In STRNN, to capture those spatially cooccurrent variations of human emotions, a multi-directional recurrent neural network (RNN) layer is employed to capture longrange contextual cues by traversing the spatial region of each time slice from multiple angles. Then a bi-directional temporal RNN layer is further used to learn discriminative temporal dependencies from the sequences concatenating spatial features of each time slice produced from the spatial RNN layer. To further select those salient regions of emotion representation, we impose sparse projection onto those hidden states of spatial and temporal domains, which actually also increases the model discriminant ability because of this global consideration. Consequently, such a two-layer RNN model builds spatial dependencies as well as temporal dependencies of the input signals. Experimental results on the public emotion datasets of EEG and facial expression demonstrate the proposed STRNN method is more competitive over those state-of-the-art methods