Abstract. This paper focuses on studying the denoising problem for positive semidefinite fourth-order tensor field estimation from noisy observations. The positive semidefiniteness of the tensor is preserved by mapping the tensor to a 6-by-6 symmetric positive semidefinite matrix where its matrix rank is less than or equal to three. For denoising, we propose to use an anisotropic discrete total variation function over the tensor field as the regularization term. We propose an inexact majorized proximal alternating direction method of multipliers for such a nonconvex and nonsmooth optimization problem. We show that an [Formula: see text]-stationary solution of the resulting optimization problem can be found in no more than [Formula: see text] iterations. The effectiveness of the proposed model and algorithm is tested using multifiber diffusion weighted imaging data, and our numerical results demonstrate that our method outperforms existing methods in terms of denoising performance.

Heterogeneous domain adaptation (HDA) is a challenging problem because of the different feature representations in the source and target domains. Most HDA methods search for mapping matrices from the source and target domains to discover latent features for learning. However, these methods barely consider the reconstruction error to measure the information loss during the mapping procedure. In this paper, we propose to jointly capture the information and match the source and target domain distributions in the latent feature space. In the learning model, we propose to minimize the reconstruction loss between the original and reconstructed representations to preserve information during transformation and reduce the Maximum Mean Discrepancy between the source and target domains to align their distributions. The resulting minimization problem involves two projection variables with orthogonal constraints that can be solved by the generalized gradient flow method, which can preserve orthogonal constraints in the computational procedure. We conduct extensive experiments on several image classification datasets to demonstrate that the effectiveness and efficiency of the proposed method are better than those of state-of-the-art HDA methods.

Although image denoising as a basic task of image restoration has been widely studied in the past decades, there are not many studies on mixed noise denoising. In this paper, we propose two structured dictionary learning models to recover images corrupted by mixed Gaussian and impulse noise. These two models can be merged as ℓp-norm fidelity plus ℓq-norm regularization. The fidelity term is used to fit image patches and the regularization term is employed for sparse coding. Particularly, we utilize proximal (and proximal linearized) alternating minimization methods as the main solvers to deal with these two models. We remove the Gaussian noise under the assumption that the uncorrupted image can be approximated with a linear representation under an appropriate orthogonal basis. We use different ways to remove impulse noise for these two models. The experimental results are reported to compare the existing methods and demonstrate the performance of the proposed denoising model is better than the other existing methods in terms of some quality assessment metrics.

We study a multi-instance (MI) learning dimensionality-reduction algorithm through sparsity and orthogonality, which is especially useful for high-dimensional MI data sets. We develop a novel algorithm to handle both sparsity and orthogonality constraints that existing methods do not handle well simultaneously. Our main idea is to formulate an optimization problem where the sparse term appears in the objective function and the orthogonality term is formed as a constraint. The resulting optimization problem can be solved by using approximate augmented Lagrangian iterations as the outer loop and inertial proximal alternating linearized minimization (iPALM) iterations as the inner loop. The main advantage of this method is that both sparsity and orthogonality can be satisfied in the proposed algorithm. We show the global convergence of the proposed iterative algorithm. We also demonstrate that the proposed algorithm can achieve high sparsity and orthogonality requirements, which are very important for dimensionality reduction. Experimental results on both synthetic and real data sets show that the proposed algorithm can obtain learning performance comparable to that of other tested MI learning algorithms.