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We introduce a new class of probabilistic cellular automata that are capable of exhibiting rich dynamics such as synchronization and ergodicity and can be easily inferred from data. The system is a finite-state locally interacting Markov chain on a circular graph. Each site's subsequent state is random, with a distribution determined by its neighborhood's empirical distribution multiplied by a local transition matrix. We establish sufficient and necessary conditions on the local transition matrix for synchronization and ergodicity. Also, we introduce novel least squares estimators for inferring the local transition matrix from various types of data, which may consist of either multiple trajectories, a long trajectory, or ensemble sequences without trajectory information. Under suitable identifiability conditions, we show the asymptotic normality of these estimators and provide non-asymptotic bounds for their accuracy., Comment: 30 pages, 3 figures

We study the community detection problem on a Gaussian mixture model, in which (1) vertices are divided into $k\geq 2$ distinct communities that are not necessarily equally-sized; (2) the Gaussian perturbations for different entries in the observation matrix are not necessarily independent or identically distributed. We prove necessary and sufficient conditions for the exact recovery of the maximum likelihood estimation (MLE), and discuss the cases when these necessary and sufficient conditions give sharp threshold. Applications include the community detection on a graph where the Gaussian perturbations of observations on each edge is the sum of i.i.d.~Gaussian random variables on its end vertices, in which we explicitly obtain the threshold for the exact recovery of the MLE., Comment: 35 pages, 4 figures. arXiv admin note: text overlap with arXiv:2009.01185

We introduce a nonlinear stochastic model reduction technique for high-dimensional stochastic dynamical systems that have a low-dimensional invariant effective manifold with slow dynamics, and high-dimensional, large fast modes. Given only access to a black box simulator from which short bursts of simulation can be obtained, we design an algorithm that outputs an estimate of the invariant manifold, a process of the effective stochastic dynamics on it, which has averaged out the fast modes, and a simulator thereof. This simulator is efficient in that it exploits of the low dimension of the invariant manifold, and takes time steps of size dependent on the regularity of the effective process, and therefore typically much larger than that of the original simulator, which had to resolve the fast modes. The algorithm and the estimation can be performed on-the-fly, leading to efficient exploration of the effective state space, without losing consistency with the underlying dynamics. This construction enables fast and efficient simulation of paths of the effective dynamics, together with estimation of crucial features and observables of such dynamics, including the stationary distribution, identification of metastable states, and residence times and transition rates between them.

In this paper, we consider the problem of super-resolution recons-truction. This is a hot topic because super-resolution reconstruction has a wide range of applications in the medical field, remote sensing monitoring, and criminal investigation. Compared with traditional algorithms, the current super-resolution reconstruction algorithm based on deep learning greatly improves the clarity of reconstructed pictures. Existing work like Super-Resolution Using a Generative Adversarial Network (SRGAN) can effectively restore the texture details of the image. However, experimentally verified that the texture details of the image recovered by the SRGAN are not robust. In order to get super-resolution reconstructed images with richer high-frequency details, we improve the network structure and propose a super-resolution reconstruction algorithm combining wavelet transform and Generative Adversarial Network. The proposed algorithm can efficiently reconstruct high-resolution images with rich global information and local texture details. We have trained our model by PyTorch framework and VOC2012 dataset, and tested it by Set5, Set14, BSD100 and Urban100 test datasets.