scholarly journals Freeness over the diagonal and outliers detection in deformed random matrices with a variance profile

2020 ◽  
Author(s):  
Jérémie Bigot ◽  
Camille Male
Keyword(s):  
Random Matrix ◽  
Free Probability ◽  
Singular Values ◽  
Low Rank ◽  
Data Matrix ◽  
Gaussian Unitary Ensemble ◽  
Deterministic Equivalent ◽  
Rank Matrix ◽  
Additive Perturbation ◽  
Variance Profile

Abstract We study the eigenvalue distribution of a Gaussian unitary ensemble (GUE) matrix with a variance profile that is perturbed by an additive random matrix that may possess spikes. Our approach is guided by Voiculescu’s notion of freeness with amalgamation over the diagonal and by the notion of deterministic equivalent. This allows to derive a fixed point equation to approximate the spectral distribution of certain deformed GUE matrices with a variance profile and to characterize the location of potential outliers in such models in a non-asymptotic setting. We also consider the singular values distribution of a rectangular Gaussian random matrix with a variance profile in a similar setting of additive perturbation. We discuss the application of this approach to the study of low-rank matrix denoising models in the presence of heteroscedastic noise, that is when the amount of variance in the observed data matrix may change from entry to entry. Numerical experiments are used to illustrate our results. Deformed random matrix, Variance profile, Outlier detection, Free probability, Freeness with amalgamation, Operator-valued Stieltjes transform, Gaussian spiked model, Low-rank model. 2000 Math Subject Classification: 62G05, 62H12.

scholarly journals On critical points of quadratic low-rank matrix optimization problems

2020 ◽  
Vol 40 (4) ◽  
pp. 2626-2651
Author(s):  
André Uschmajew ◽  
Bart Vandereycken
Keyword(s):  
Critical Points ◽  
Optimization Problems ◽  
Quadratic Function ◽  
Singular Values ◽  
Positive Semidefinite ◽  
Optimization Methods ◽  
Low Rank ◽  
Local Minima ◽  
Rank Matrix ◽  
Low Rank Matrix

Abstract The absence of spurious local minima in certain nonconvex low-rank matrix recovery problems has been of recent interest in computer science, machine learning and compressed sensing since it explains the convergence of some low-rank optimization methods to global optima. One such example is low-rank matrix sensing under restricted isometry properties (RIPs). It can be formulated as a minimization problem for a quadratic function on the Riemannian manifold of low-rank matrices, with a positive semidefinite Riemannian Hessian that acts almost like an identity on low-rank matrices. In this work new estimates for singular values of local minima for such problems are given, which lead to improved bounds on RIP constants to ensure absence of nonoptimal local minima and sufficiently negative curvature at all other critical points. A geometric viewpoint is taken, which is inspired by the fact that the Euclidean distance function to a rank-$k$ matrix possesses no critical points on the corresponding embedded submanifold of rank-$k$ matrices except for the single global minimum.

scholarly journals FREE DETERMINISTIC EQUIVALENTS, RECTANGULAR RANDOM MATRIX MODELS, AND OPERATOR-VALUED FREE PROBABILITY THEORY

2012 ◽  
Vol 01 (02) ◽  
pp. 1150008 ◽  
Author(s):  
ROLAND SPEICHER ◽  
CARLOS VARGAS
Keyword(s):  
Matrix Models ◽  
Random Matrix ◽  
Collective Behavior ◽  
Free Probability ◽  
Matrix Problem ◽  
Free Probability Theory ◽  
Random Matrix Model ◽  
Deterministic Equivalent ◽  
Cauchy Transforms ◽  
Random Matrix Models

Motivated by the asymptotic collective behavior of random and deterministic matrices, we propose an approximation (called "free deterministic equivalent") to quite general random matrix models, by replacing the matrices with operators satisfying certain freeness relations. We comment on the relation between our free deterministic equivalent and deterministic equivalents considered in the engineering literature. We do not only consider the case of square matrices, but also show how rectangular matrices can be treated. Furthermore, we emphasize how operator-valued free probability techniques can be used to solve our free deterministic equivalents. As an illustration of our methods we show how the free deterministic equivalent of a random matrix model from [6] can be treated and we thus recover in a conceptual way the results from [6]. On a technical level, we generalize a result from scalar-valued free probability, by showing that randomly rotated deterministic matrices of different sizes are asymptotically free from deterministic rectangular matrices, with amalgamation over a certain algebra of projections. In Appendix A, we show how estimates for differences between Cauchy transforms can be extended from a neighborhood of infinity to a region close to the real axis. This is of some relevance if one wants to compare the original random matrix problem with its free deterministic equivalent.

Low-rank matrix decomposition method for potential field data separation

Geophysics ◽  
2020 ◽  
Vol 85 (1) ◽  
pp. G1-G16
Author(s):  
Dan Zhu ◽  
Hongwei Li ◽  
Tianyou Liu ◽  
Lihua Fu ◽  
Shihui Zhang
Keyword(s):  
Potential Field ◽  
Field Data ◽  
Matrix Decomposition ◽  
Low Rank ◽  
Data Matrix ◽  
Potential Field Data ◽  
Residual Field ◽  
Optimization Task ◽  
Rank Matrix ◽  
Low Rank Matrix

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.

scholarly journals Fabric defect detection based on deep-handcrafted feature and weighted low-rank matrix representation

2021 ◽  
Vol 16 ◽  
pp. 155892502110084
Author(s):  
Chunlei Li ◽  
Ban Jiang ◽  
Zhoufeng Liu ◽  
Yan Dong ◽  
Shuili Tang ◽  
...  
Keyword(s):  
Defect Detection ◽  
Matrix Representation ◽  
Singular Values ◽  
Low Rank ◽  
Fabric Defect Detection ◽  
Rank Matrix ◽  
Fabric Defect ◽  
The Matrix ◽  
Low Rank Representation ◽  
Low Rank Matrix

In the process of textile production, automatic defect detection plays a key role in controlling product quality. Due to the complex texture features of fabric image, the traditional detection methods have poor adaptability, and low detection accuracy. The low rank representation model can divide the image into the low rank background and sparse object, and has proven suitable for fabric defect detection. However, how to further effectively characterize the fabric texture is still problematic in this kind of method. Moreover, most of them adopt nuclear norm optimization algorithm to solve the low rank model, which treat every singular value in the matrix equally. However, in the task of fabric defect detection, different singular values of feature matrix represent different information. In this paper, we proposed a novel fabric defect detection method based on the deep-handcrafted feature and weighted low-rank matrix representation. The feature characterization ability is effectively improved by fusing the global deep feature extracted by VGG network and the handcrafted low-level feature. Moreover, a weighted low-rank representation model is constructed to treat the matrix singular values differently by different weights, thus the most distinguishing feature of fabric texture can be preserved, which can efficiently outstand the defect and suppress the background. Qualitative and quantitative experiments on two public datasets show that our proposed method outperforms the state-of-the-art methods.

scholarly journals Robust and Low-Complexity Cooperative Spectrum Sensing via Low-Rank Matrix Recovery in Cognitive Vehicular Networks

2018 ◽  
Vol 2018 ◽  
pp. 1-14
Author(s):  
Xia Liu ◽  
Zhimin Zeng ◽  
Caili Guo
Keyword(s):  
Spectrum Sensing ◽  
Vehicular Networks ◽  
Cooperative Spectrum Sensing ◽  
Low Rank ◽  
Data Matrix ◽  
Sensing Data ◽  
Matrix Recovery ◽  
Rank Matrix ◽  
Low Rank Matrix Recovery ◽  
Low Rank Matrix

In cognitive vehicular networks (CVNs), many envisioned applications related to safety require highly reliable connectivity. This paper investigates the issue of robust and efficient cooperative spectrum sensing in CVNs. We propose robust cooperative spectrum sensing via low-rank matrix recovery (LRMR-RCSS) in cognitive vehicular networks to address the uncertainty of the quality of potentially corrupted sensing data by utilizing the real spectrum occupancy matrix and corrupted data matrix, which have a simultaneously low-rank and joint-sparse structure. Considering that the sensing data from crowd cognitive vehicles would be vast, we extend our robust cooperative spectrum sensing algorithm to dense cognitive vehicular networks via weighted low-rank matrix recovery (WLRMR-RCSS) to reduce the complexity of cooperative spectrum sensing. In the WLRMR-RCSS algorithm, we propose a correlation-aware selection and weight assignment scheme to take advantage of secondary user (SU) diversity and reduce the cooperation overhead. Extensive simulation results demonstrate that the proposed LRMR-RCSS and WLRMR-RCSS algorithms have good performance in resisting malicious SU behavior. Moreover, the simulations demonstrate that the proposed WLRMR-RCSS algorithm could be successfully applied to a dense traffic environment.

Optimized quantum singular value thresholding algorithm based on a hybrid quantum computer

2021 ◽  
Author(s):  
Yangyang Ge ◽  
Zhimin Wang ◽  
Wen Zheng ◽  
Yu Zhang ◽  
Xiangmin Yu ◽  
...  
Keyword(s):  
Quantum Computer ◽  
Singular Values ◽  
Singular Value ◽  
Low Rank ◽  
Continuous Variables ◽  
Rank Matrix ◽  
Near Term ◽  
Thresholding Algorithm ◽  
Low Rank Matrix ◽  
Singular Value Thresholding

Abstract Quantum singular value thresholding (QSVT) algorithm, as a core module of many mathematical models, seeks the singular values of a sparse and low rank matrix exceeding a threshold and their associated singular vectors. The existing all-qubit QSVT algorithm demands lots of ancillary qubits, remaining a huge challenge for realization on near-term intermediate-scale quantum computers. In this paper, we propose a hybrid QSVT (HQSVT) algorithm utilizing both discrete variables (DVs) and continuous variables (CVs). In our algorithm, raw data vectors are encoded into a qubit system and the following data processing is fulfilled by hybrid quantum operations. Our algorithm requires O[log(MN)] qubits with O(1) qumodes and totally performs O(1) operations, which significantly reduces the space and runtime consumption.

scholarly journals LogDet Rank Minimization with Application to Subspace Clustering

2015 ◽  
Vol 2015 ◽  
pp. 1-10 ◽  
Author(s):  
Zhao Kang ◽  
Chong Peng ◽  
Jie Cheng ◽  
Qiang Cheng
Keyword(s):  
Large Scale ◽  
Clustering Algorithms ◽  
Subspace Clustering ◽  
Singular Values ◽  
Nuclear Norm ◽  
Low Rank ◽  
Face Clustering ◽  
Rank Matrix ◽  
Low Rank Representation ◽  
Clustering Data

Low-rank matrix is desired in many machine learning and computer vision problems. Most of the recent studies use the nuclear norm as a convex surrogate of the rank operator. However, all singular values are simply added together by the nuclear norm, and thus the rank may not be well approximated in practical problems. In this paper, we propose using a log-determinant (LogDet) function as a smooth and closer, though nonconvex, approximation to rank for obtaining a low-rank representation in subspace clustering. Augmented Lagrange multipliers strategy is applied to iteratively optimize the LogDet-based nonconvex objective function on potentially large-scale data. By making use of the angular information of principal directions of the resultant low-rank representation, an affinity graph matrix is constructed for spectral clustering. Experimental results on motion segmentation and face clustering data demonstrate that the proposed method often outperforms state-of-the-art subspace clustering algorithms.

scholarly journals Image Denoising Using Low Rank Matrix Approximation in Singular Value Decomposition

2021 ◽  
Vol 11 (2) ◽  
pp. 1430-1446
Author(s):  
Satyanarayana Tallapragada V.V.
Keyword(s):  
Singular Value Decomposition ◽  
Image Denoising ◽  
Singular Values ◽  
Singular Value ◽  
Low Rank ◽  
Lower Rank ◽  
Rank Matrix ◽  
Wide Range ◽  
Value Decomposition ◽  
Low Rank Matrix

The factorization of a matrix into lower rank matrices give solutions to a wide range of computer vision and image processing tasks. The inherent patches or the atomic patches can completely describe the whole image. The lower rank matrices are obtained using different tools including Singular Value Decomposition (SVD), which is typically found in minimization problems of nuclear norms. The singular values obtained will generally be a thresholder to realize the nuclear norm minimization. However, soft-thresholding is performed uniformly on all the singular values that lead to a similar importance to all the patches whether it is principal/useful or not. Our observation is that the decision on a patch (to be principal/useful or not) can be taken only when the application of this minimization is taken into consideration. Thus, in this paper, we propose a new method for image denoising by choosing variable weights to different singular values with a deep noise effect. Experimental results illustrate that the proposed weighted scheme performs better than the state-of-the-art methods.

Sign in / Sign up

Export Citation Format

Share Document