Optimal correction of infeasible equations system as Ax + B|x|= b using ℓ p-norm regularization

Optimal correction of an infeasible equations system as Ax + B|x|= b leads into a non-convex fractional problem. In this paper, a regularization method(ℓp-norm, 0 < p < 1), is presented to solve mentioned fractional problem. In this method, the obtained problem can be formulated as a non-convex and nonsmooth optimization problem which is not Lipschitz. The objective function of this problem can be decomposed as a difference of convex functions (DC). For this reason, we use a special smoothing technique based on DC programming. The numerical results obtained for generated problem show high performance and the effectiveness of the proposed method.

Efficient Iterative Regularization Method for Total Variation-Based Image Restoration

Total variation (TV) regularization has received much attention in image restoration applications because of its advantages in denoising and preserving details. A common approach to address TV-based image restoration is to design a specific algorithm for solving typical cost function, which consists of conventional ℓ2 fidelity term and TV regularization. In this work, a novel objective function and an efficient algorithm are proposed. Firstly, a pseudoinverse transform-based fidelity term is imposed on TV regularization, and a closely-related optimization problem is established. Then, the split Bregman framework is used to decouple the complex inverse problem into subproblems to reduce computational complexity. Finally, numerical experiments show that the proposed method can obtain satisfactory restoration results with fewer iterations. Combined with the restoration effect and efficiency, this method is superior to the competitive algorithm. Significantly, the proposed method has the advantage of a simple solving structure, which can be easily extended to other image processing applications.

Smooth Group L1/2 Regularization for Pruning Convolutional Neural Networks

In this paper, a novel smooth group L1/2 (SGL1/2) regularization method is proposed for pruning hidden nodes of the fully connected layer in convolution neural networks. Usually, the selection of nodes and weights is based on experience, and the convolution filter is symmetric in the convolution neural network. The main contribution of SGL1/2 is to try to approximate the weights to 0 at the group level. Therefore, we will be able to prune the hidden node if the corresponding weights are all close to 0. Furthermore, the feasibility analysis of this new method is carried out under some reasonable assumptions due to the smooth function. The numerical results demonstrate the superiority of the SGL1/2 method with respect to sparsity, without damaging the classification performance.

A high accurate method for solving an inverse problem of the Laplace equation in detection of a Robin coefficient

This study This work deals with an inverse problem for the harmonic equation to recover a Robin coefficient on a non-accessible part of a circle from Cauchy data measured on an accessible part of that circle. By assuming that the available data has a Fourier expansion, we adopt the Modified Collocation Trefftz Method (MCTM) to solve this problem. We use the truncation regularization method in combination with the collocation technique to approximate the solution, and the conjugate gradient method to obtain the coefficients, thus completing the missing Cauchy data. We recommend the least squares method to achieve a better stability. Finally, we illustrate the feasibility of this method with numerical examples.

Diffusion tensor regularization with metric double integrals

In this paper, we propose a variational regularization method for denoising and inpainting of diffusion tensor magnetic resonance images. We consider these images as manifold-valued Sobolev functions, i.e. in an infinite dimensional setting, which are defined appropriately. The regularization functionals are defined as double integrals, which are equivalent to Sobolev semi-norms in the Euclidean setting. We extend the analysis of [14] concerning stability and convergence of the variational regularization methods by a uniqueness result, apply them to diffusion tensor processing, and validate our model in numerical examples with synthetic and real data.

On the benefits of representation regularization in invariance based domain generalization

A crucial aspect of reliable machine learning is to design a deployable system for generalizing new related but unobserved environments. Domain generalization aims to alleviate such a prediction gap between the observed and unseen environments. Previous approaches commonly incorporated learning the invariant representation for achieving good empirical performance. In this paper, we reveal that merely learning the invariant representation is vulnerable to the related unseen environment. To this end, we derive a novel theoretical analysis to control the unseen test environment error in the representation learning, which highlights the importance of controlling the smoothness of representation. In practice, our analysis further inspires an efficient regularization method to improve the robustness in domain generalization. The proposed regularization is orthogonal to and can be straightforwardly adopted in existing domain generalization algorithms that ensure invariant representation learning. Empirical results show that our algorithm outperforms the base versions in various datasets and invariance criteria.

Asymptotics solutions of a singularly perturbed integro-differential fractional order derivative equation with rapidly oscillating coefficients

In this paper, the regularization method of S.A.Lomov is generalized to the singularly perturbed integrodifferential fractional-order derivative equation with rapidly oscillating coefficients. The main goal of the work is to reveal the influence of the oscillating components on the structure of the asymptotics of the solution to this problem. The case of the absence of resonance is considered, i.e. the case when an integer linear combination of a rapidly oscillating inhomogeneity does not coincide with a point in the spectrum of the limiting operator at all points of the considered time interval. The case of coincidence of the frequency of a rapidly oscillating inhomogeneity with a point in the spectrum of the limiting operator is called the resonance case. This case is supposed to be studied in our subsequent works. More complex cases of resonance (for example, point resonance) require more careful analysis and are not considered in this work.