Matrix Nuclear Norm Research Articles

Nonlocal image denoising via adaptive tensor nuclear norm minimization

Nonlocal self-similarity shows great potential in image denoising. Therefore, the denoising performance can be attained by accurately exploiting the nonlocal prior. In this paper, we model nonlocal similar patches through the multi-linear approach and then propose two tensor-based methods for image denoising. Our methods are based on the study of low-rank tensor estimation (LRTE). By exploiting low-rank prior in the tensor presentation of similar patches, we devise two new adaptive tensor nuclear norms (i.e., ATNN-1 and ATNN-2) for the LRTE problem. Among them, ATNN-1 relaxes the general tensor N-rank in a weighting scheme, while ATNN-2 is defined based on a novel tensor singular-value decomposition (t-SVD). Both ATNN-1 and ATNN-2 construct the stronger spatial relationship between patches than the matrix nuclear norm. Regularized by ATNN-1 and ATNN-2 respectively, the derived two LRTE algorithms are implemented through the adaptive singular-value thresholding with global optimal guarantee. Then, we embed the two algorithms into a residual-based iterative framework to perform nonlocal image denoising. Experiments validate the rationality of our tensor low-rank assumption, and the denoising results demonstrate that our proposed two methods are exceeding the state-of-the-art methods, both visually and quantitatively.

Learning Local Feature Descriptors Using Convex Optimisation.

The objective of this work is to learn descriptors suitable for the sparse feature detectors used in viewpoint invariant matching. We make a number of novel contributions towards this goal. First, it is shown that learning the pooling regions for the descriptor can be formulated as a convex optimisation problem selecting the regions using sparsity. Second, it is shown that descriptor dimensionality reduction can also be formulated as a convex optimisation problem, using Mahalanobis matrix nuclear norm regularisation. Both formulations are based on discriminative large margin learning constraints. As the third contribution, we evaluate the performance of the compressed descriptors, obtained from the learnt real-valued descriptors by binarisation. Finally, we propose an extension of our learning formulations to a weakly supervised case, which allows us to learn the descriptors from unannotated image collections. It is demonstrated that the new learning methods improve over the state of the art in descriptor learning on the annotated local patches data set of Brown et al. and unannotated photo collections of Philbin et al.

Recovering low-rank matrices from corrupted observations via the linear conjugate gradient algorithm

The matrix nuclear norm minimization problem has received much attention in recent years, largely because its highly related to the matrix rank minimization problem arising from controller design, signal processing and model reduction. The alternating direction method is a very popular way to solve this problem due to its simplicity, low storage, practical computation efficiency and nice convergence properties. In this paper, we propose an alternating direction method, where one variable is determined explicitly, and the other variable is computed by a linear conjugate gradient algorithm. At each iteration, the method involves a singular value thresholding and its convergence result is guaranteed in this literature. Extensive experiments illustrate that the proposed algorithm compares favorable with the state-of-the-art algorithms FPCA and IADM_BB which were specifically designed in recent years.

An Efficient Matrix Factorization Method for Tensor Completion

Most recent low-rank tensor completion algorithms are based on tensor nuclear norm minimization problems. The convex relaxation problem of tensor <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> -rank minimization has to be solved iteratively and involves multiple singular value decompositions (SVDs) at each iteration, and thus such algorithms suffer from high computation cost. In this letter, we propose an efficient low-rank tensor completion approach. First, we introduce a matrix factorization idea into the tensor nuclear norm model, and then can achieve a much smaller scale matrix nuclear norm minimization problem. Moreover, we develop an efficient iterative scheme for solving the proposed model with orthogonality constraint. Our extensive evaluation results validate both the effectiveness and efficiency of the proposed approach.

An efficient matrix factorization based low-rank representation for subspace clustering

In recent years, robust subspace clustering is an important unsupervised clustering problem in machine learning and computer vision communities. The recently proposed spectral clustering based approach, called low-rank representation (LRR), yields an optimal solution for the case of independent subspaces and partially corrupted data. However, it has to be solved iteratively and involves singular value decomposition (SVD) at each iteration, and then suffers from high computation cost of multiple SVDs. In this paper, we propose an efficient matrix tri-factorization (MTF) approach with a positive semidefinite (PSD) constraint to approximate the original nuclear norm minimization (NNM) problem and mitigate the computation cost of performing SVDs. Specially, we introduce a matrix tri-factorization idea into the original low-rank representation framework, and then convert it into a small scale matrix nuclear norm minimization problem. Finally, we establish an alternating direction method (ADM) based algorithm to efficiently solve the proposed problem. Experimental results on a variety of synthetic and real-world data sets validate the efficiency, robustness and effectiveness of the proposed MTF approach comparing with the state-of-the-art algorithms.

An alternating direction method for linear‐constrained matrix nuclear norm minimization

SUMMARYThe aim of the nuclear norm minimization problem is to find a matrix that minimizes the sum of its singular values and satisfies some constraints simultaneously. Such a problem has received more attention largely because it is closely related to the affine rank minimization problem, which appears in many control applications including controller design, realization theory, and model reduction. In this paper, we first propose an exact version alternating direction method for solving the nuclear norm minimization problem with linear equality constraints. At each iteration, the method involves a singular value thresholding and linear matrix equations which are solved exactly. Convergence of the proposed algorithm is followed directly. To broaden the capacity of solving larger problems, we solve approximately the subproblem by an iterative method with the Barzilai–Borwein steplength. Some extensions to the noisy problems and nuclear norm regularized least‐square problems are also discussed. Numerical experiments and comparisons with the state‐of‐the‐art method FPCA show that the proposed method is effective and promising. Copyright © 2011 John Wiley &amp; Sons, Ltd.