Inexact Augmented Lagrange Multiplier Algorithm Research Articles

Fast non-convex low-rank matrix decomposition for separation of potential field data using minimal memory

A fast non-convex low-rank matrix decomposition method for potential field data separation is proposed. The singular value decomposition of the large size trajectory matrix, which is also a block Hankel matrix, is obtained using a fast randomized singular value decomposition algorithm in which fast block Hankel matrix-vector multiplications are implemented with minimal memory storage. This fast block Hankel matrix randomized singular value decomposition algorithm is integrated into the \texttt{Altproj} algorithm, which is a standard non-convex method for solving the robust principal component analysis optimization problem. The improved algorithm avoids the construction of the trajectory matrix. Hence, gravity and magnetic data matrices of large size can be computed. Moreover, it is more efficient than the traditional low-rank matrix decomposition method, which is based on the use of an inexact augmented Lagrange multiplier algorithm. The presented algorithm is also robust and, hence, algorithm-dependent parameters are easily determined. The improved and traditional algorithms are contrasted for the separation of synthetic gravity and magnetic data matrices of different sizes. The presented results demonstrate that the improved algorithm is not only computationally more efficient but it is also more accurate. Moreover, it is possible to solve far larger problems. As an example, for the adopted computational environment, matrices of sizes larger than $205 \times 205$ generate "out of memory" exceptions with the traditional method, but a matrix of size $2001\times 2001$ can be calculated in $1062.29$s with the new algorithm. Finally, the improved method is applied to separate real gravity and magnetic data in the Tongling area, Anhui province, China. Areas which may exhibit mineralizations are inferred based on the separated anomalies.

Fast and Latent Low-Rank Subspace Clustering for Hyperspectral Band Selection

This article presents a fast and latent low-rank subspace clustering (FLLRSC) method to select hyperspectral bands. The FLLRSC assumes that all the bands are sampled from a union of latent low-rank independent subspaces and formulates the self-representation property of all bands into a latent low-rank representation (LLRR) model. The assumption ensures sufficient sampling bands in representing low-rank subspaces of all bands and improves robustness to noise. The FLLRSC first implements the Hadamard random projections to reduce spatial dimensionality and lower the computational cost. It then adopts the inexact augmented Lagrange multiplier algorithm to optimize the LLRR program and estimates sparse coefficients of all the projected bands. After that, it employs a correntropy metric to measure the similarity between pairwise bands and constructs an affinity matrix based on sparse representation. The correntropy metric could better describe the nonlinear characteristics of hyperspectral bands and enhance the block-diagonal structure of the similarity matrix for correctly clustering all subspaces. The FLLRSC conducts spectral clustering on the connected graph denoted by the affinity matrix. The bands that are closest to their separate cluster centroids form the final band subset. Experimental results on three widely used hyperspectral data sets show that the FLLRSC performs better than the classical low-rank representation methods with higher classification accuracy at a low computational cost.

Multi-Scale Infrared Small Target Detection Method via Precise Feature Matching and Scale Selection Strategy

Infrared small target detection is a crucial and challenging topic for various applications. In recent years, the spectrum scale space (SSS) algorithm has shown considerable potential in the field of target detection. However, the SSS algorithm is prone to high false alarm rates in infrared small target detection scenarios with complex background. This paper proposes an improved SSS (ISSS) algorithm via precise feature matching and scale selection strategy for efficient infrared small target detection, which includes background suppression, feature matching and optimal scale selection three stages. In the background suppression stage, a matrix decomposition method named inexact augmented Lagrange multiplier (IALM) algorithm is used to extract the sparse image matrix from the original image as the target foreground image. In the feature matching stage, the 16 elaborate Gaussian kernel functions convolve with the the amplitude spectrum of target foreground image to generate 16 scale saliency maps that precisely match the feature of small targets. In the optimal scale selection stage, a few proper candidate scale maps are screened out according to the difference between the pixel values of the target area and the background clutters, in which the target area was more highlighted, and the scale map corresponding to the maximum value of local information entropy of the candidate saliency map is the final detection result map. We mainly made three contributions: First, IALM algorithm is utilized as a preprocessing step, and we have verified it is indispensable in eliminating most backgrounds with self-correlation property. Second, an elaborate scale division strategy is proposed to obtain multi-scale saliency maps that match the feature of infrared small targets precisely. Third, the gray value difference and the maximum value of local information entropy are defined and used as the judgment criteria for optimal scale selection. Extensive experimental results demonstrate that the proposed method outperforms state-of-the-art techniques, especially on infrared images with thick clouds and high-brightness buildings.

Low-rank matrix decomposition method for potential field data separation

Separation of potential field data forms the basis of inversion and interpretation. The low-rank matrix theory is used for the separation of potential field data. A theoretical analysis led to the approximate equation that demonstrates the relation between the amplitudes of the wavenumber components of potential field data and the singular values of the trajectory matrix embedded from the potential field data matrix. Therefore, the low-rank feature of the trajectory matrix of regional field data and the sparse feature of the trajectory matrix of residual field data can be obtained based on the features of the power spectrum of the potential field data. Based on this, we have developed a low-rank matrix decomposition model for the separation of the trajectory matrix of the potential field data. Minimizing the rank of the trajectory matrix of the regional field data and the [Formula: see text]-norm of the trajectory matrix of the residual field data is a double-objective optimization task, and this optimization task can be solved by the inexact augmented Lagrange multiplier algorithm. The weighting parameter is robust and easy to set. Numerical experiment results indicate that our method is robust, and the separation errors of the method are smaller compared to the matched filtering and wavelet analysis methods. In addition, our method can be easily implemented and has clear geophysical significance. Finally, our method is applied on real data sets in the Daye area, Hubei Province, China. The separated gravity and magnetic fields coincide well with target geologic sources.

Constrained Low-Rank Learning Using Least Squares-Based Regularization.

Low-rank learning has attracted much attention recently due to its efficacy in a rich variety of real-world tasks, e.g., subspace segmentation and image categorization. Most low-rank methods are incapable of capturing low-dimensional subspace for supervised learning tasks, e.g., classification and regression. This paper aims to learn both the discriminant low-rank representation (LRR) and the robust projecting subspace in a supervised manner. To achieve this goal, we cast the problem into a constrained rank minimization framework by adopting the least squares regularization. Naturally, the data label structure tends to resemble that of the corresponding low-dimensional representation, which is derived from the robust subspace projection of clean data by low-rank learning. Moreover, the low-dimensional representation of original data can be paired with some informative structure by imposing an appropriate constraint, e.g., Laplacian regularizer. Therefore, we propose a novel constrained LRR method. The objective function is formulated as a constrained nuclear norm minimization problem, which can be solved by the inexact augmented Lagrange multiplier algorithm. Extensive experiments on image classification, human pose estimation, and robust face recovery have confirmed the superiority of our method.

Complex and Quaternionic Principal Component Pursuit and Its Application to Audio Separation

Recently, the principal component pursuit has received increasing attention in signal processing research ranging from source separation to video surveillance. So far, all existing formulations are real-valued and lack the concept of phase, which is inherent in inputs such as complex spectrograms or color images. Thus, in this letter, we extend principal component pursuit to the complex and quaternionic cases to account for the missing phase information. Specifically, we present both complex and quaternionic proximity operators for the $\ell_1$- and trace-norm regularizers. These operators can be used in conjunction with proximal minimization methods such as the inexact augmented Lagrange multiplier algorithm. The new algorithms are then applied to the singing voice separation problem, which aims to separate the singing voice from the instrumental accompaniment. Results on the iKala and MSD100 datasets confirmed the usefulness of phase information in principal component pursuit.

Robust Subspace Segmentation Via Low-Rank Representation

Recently the low-rank representation (LRR) has been successfully used in exploring the multiple subspace structures of data. It assumes that the observed data is drawn from several low-rank subspaces and sometimes contaminated by outliers and occlusions. However, the noise (low-rank representation residual) is assumed to be sparse, which is generally characterized by minimizing the l1 -norm of the residual. This actually assumes that the residual follows the Laplacian distribution. The Laplacian assumption, however, may not be accurate enough to describe various noises in real scenarios. In this paper, we propose a new framework, termed robust low-rank representation, by considering the low-rank representation as a low-rank constrained estimation for the errors in the observed data. This framework aims to find the maximum likelihood estimation solution of the low-rank representation residuals. We present an efficient iteratively reweighted inexact augmented Lagrange multiplier algorithm to solve the new problem. Extensive experimental results show that our framework is more robust to various noises (illumination, occlusion, etc) than LRR, and also outperforms other state-of-the-art methods.