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This paper proposes a novel deep learning (DL) based source reconstruction method (SRM). The proposed DL-based SRM employs the deep convolutional asymmetric encoder–decoder structure (DCAEDS), which only demands one-time single-frequency far-field measurement on EM scattered field as its input and further predicts the equivalent source on target scatterers. During the offline training, an EM scattering simulator is designed to compute EM scattered field originated from the predicted equivalent source on target scatterers. The DCAEDS is combined with the EM scattering simulator to optimize the loss function, including two parts: (1) Data-induced loss, which directly evaluates the difference between the prediction of the proposed DCAEDS and the true-labelled equivalent source of target scatterers; (2) Physics-induced loss, which evaluates the difference between received EM scattered field and the computed EM scattered field originated from the prediction of DCAEDS. Moreover, the proposed DL-based SRM can overcome the limitation of conventional methods, involving high computation cost and strong ill-conditions. Consequently, the proposed DL-based SRM can realize the reconstruction of the equivalent current source with higher accuracy and lower computation complexity. Numerical examples illustrate the feasibility of the proposed DL-based SRM, which opens the new path for DL-based EM computation approaches. [ABSTRACT FROM AUTHOR]

We study a class of spectral learning methods with dependent observations, including popular ridge regression, Landweber iteration, spectral cut-off and so on. We derive an explicit risk bound in terms of the correlation of the observations, regularity of the regression function, and effective dimension of the reproducing kernel Hilbert space. By appropriately choosing regularization parameter according to the sample size, the risk bound yields a nearly optimal learning rate with a logarithmic term for strongly mixing sequences. We thus extend the applicable range of spectral algorithm to non-i.i.d. sampling process. Particularly, it is shown that the learning rates for i.i.d. samples in the literature refer to our special case, i.e., the mixing condition parameter tends to zero. [ABSTRACT FROM AUTHOR]

The goal of tensor completion is to fill in missing entries of a partially known tensor under a low-rank constraint. In this paper, we study low rank third-order tensor completion problems by using Riemannian optimization methods on the smooth manifold. Here the tensor rank is defined to be a multi-rank under the Discrete Cosine Transform-related transform tensor-tensor product. With suitable incoherence conditions on the underlying low multi-rank rank tensor, we show that the proposed Riemannian optimization method is guaranteed to converge to the underlying low multi-rank tensor with a high probability. Number of sampling entries required for convergence are also derived. Numerical examples of synthetic data and real data sets are reported to demonstrate that the performance of the proposed method is better than that of tensor-based method using the Tucker-rank model in terms of computational time, and that of the matrix-based completion method in terms of number of sampling entries. [ABSTRACT FROM AUTHOR]