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In this paper, we study deep neural networks (DNNs) for solving high-dimensional evolution equations with oscillatory solutions. Different from deep least-squares methods that deal with time and space variables simultaneously, we propose a deep adaptive basis Galerkin (DABG) method which employs the spectral-Galerkin method for time variable by a tensor-product basis for oscillatory solutions and the deep neural network method for high-dimensional space variables. The proposed method can lead to a linear system of differential equations having unknown DNNs that can be trained via the loss function. We establish a posteriori estimates of the solution error which is bounded by the minimal loss function and the term O(N-m), where N is the number of basis functions and m characterizes the regularity of the equation, and show that if the true solution is a Barron-type function, the error bound converges to zero as M = O(Np) approaches to infinity where M is the width of the used networks and p is a positive constant. Numerical examples, including high-dimensional linear parabolic and hyperbolic equations, and a nonlinear Allen--Cahn equation are presented to demonstrate that the performance of the proposed DABG method is better than that of existing DNNs. [ABSTRACT FROM AUTHOR]

Poisson observations for videos are important models in video processing and computer vision. In this paper, we study the third-order tensor completion problem with Poisson observations. The main aim is to recover a tensor based on a small number of its Poisson observation entries. A existing matrix-based method may be applied to this problem via the matricized version of the tensor. However, this method does not leverage on the global low-rankness of a tensor and may be substantially suboptimal. Our approach is to consider the maximum likelihood estimate of the Poisson distribution, and utilize the Kullback-Leibler divergence for the data-fitting term to measure the observations and the underlying tensor. Moreover, we propose to employ a transformed tensor nuclear norm ball constraint and a bounded constraint of each entry, where the transformed tensor nuclear norm is used to get a lower transformed multi-rank tensor with suitable unitary transformation matrices. We show that the upper bound of the error of the estimator of the proposed model is less than that of the existing matrix-based method. Also an information theoretic lower error bound is established. An alternating direction method of multipliers is developed to solve the resulting convex optimization model. Extensive numerical experiments on synthetic data and real-world datasets are presented to demonstrate the effectiveness of our proposed model compared with existing tensor completion methods. [ABSTRACT FROM AUTHOR]

In this paper, we propose a novel method for solving high-dimensional spectral fractional Laplacian equations. Using the Caffarelli--Silvestre extension, the d-dimensional spectral fractional equation is reformulated as a regular partial differential equation of dimension d+1. We transform the extended equation as a minimal Ritz energy functional problem and search for its minimizer in a special class of deep neural networks. Moreover, based on the approximation property of networks, we establish estimates on the error made by the deep Ritz method. Numerical results are reported to demonstrate the effectiveness of the proposed method for solving fractional Laplacian equations up to 10 dimensions. Technically, in this method, we design a special network-based structure to adapt to the singularity and exponential decaying of the true solution. Also, a hybrid integration technique combining the Monte Carlo method and sinc quadrature is developed to compute the loss function with higher accuracy. [ABSTRACT FROM AUTHOR]

Transformed L1 (TL1) regularization has been shown to have comparable signal recovery capability with L1 L2 regularization and L1/L2 regularization, regardless of whether the measurement matrix satisfies the restricted isometry property (RIP). In the spirit of the TL1 method, we introduce a transformed total variation (TTV) minimization model to investigate robust image recovery from a certain number of noisy measurements by the proposed TTV minimization model in this paper. An optimal error bound, up to a logarithmic factor, of robust image recovery from compressed measurements via the TTV minimization model is established, and the RIP based condition is improved compared with total variation (TV) minimization. Numerical results of image reconstruction demonstrate our theoretical results and illustrate the efficiency of the TTV minimization model among state-of-the-art methods. Empirically, the error bound between the reconstructed image and the original image is shown to be better than that produced by TV minimization. [ABSTRACT FROM AUTHOR]

In this paper, we study color image inpainting as a pure quaternion matrix completion problem. In the literature, the theoretical guarantee for quaternion matrix completion is not well established. Our main aim is to propose a new minimization problem with an objective combining nuclear norm and a quadratic loss weighted among three channels. To fill the theoretical vacancy, we obtain the error bound in both clean and corrupted regimes, which relies on some new results of quaternion matrices. A general Gaussian noise is considered in robust completion where all observations are corrupted. Motivated by the error bound, we propose to handle unbalanced or correlated noise via a cross-channel weight in the quadratic loss, with the main purpose of rebalancing noise level or removing noise correlation. Extensive experimental results on synthetic and color image data are presented to confirm and demonstrate our theoretical findings. [ABSTRACT FROM AUTHOR]

Color image enhancement is one of the most important tasks in image processing. In this paper, we focus on the red and/or blue attenuation problem that arises in underwater image enhancement. We propose and develop a novel spatial color compensation model based on saturation-value total variation (SV-TV). In the proposed model, the compensation of the red and/or blue attenuation is achieved by adding a fraction of the green channel to the red and/or blue channel. The fraction is determined by a spatially varying function, which is controlled by the TV regularization. In order to avoid overcompensation and too many color artifacts, we make use of the SV-TV to regularize the objective compensation result. We numerically apply the proximal alternating linearized minimization to solve the proposed minimization problem, and we give the convergence analysis of the proposed algorithm. Numerical examples are presented to demonstrate that the performance of the proposed color compensation model is better than that of other testing methods in terms of visual quality and certain criteria, such as peak signal-to-noise ratio (PSNR), structure similarity (SSIM), and CIEde2000 color difference. [ABSTRACT FROM AUTHOR]

One of the main challenges in multi-source domain adaptation is how to reduce the domain discrepancy between each source domain and a target domain, and then evaluate the domain relevance to determine how much knowledge should be transferred from different source domains to the target domain. However, most prior approaches barely consider both discrepancies and relevance among domains. In this paper, we propose an algorithm, called Iterative Refinement based on Feature Selection and the Wasserstein distance (IRFSW), to solve semi-supervised domain adaptation with multiple sources. Specifically, IRFSW aims to explore both the discrepancies and relevance among domains in an iterative learning procedure, which gradually refines the learning performance until the algorithm stops. In each iteration, for each source domain and the target domain, we develop a sparse model to select features in which the domain discrepancy and training loss are reduced simultaneously. Then a classifier is constructed with the selected features of the source and labeled target data. After that, we exploit optimal transport over the selected features to calculate the transferred weights. The weight values are taken as the ensemble weights to combine the learned classifiers to control the amount of knowledge transferred from source domains to the target domain. Experimental results validate the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]