Gaussian Random Matrix Research Articles

Random Frequency Division Multiplexing

In this paper, we propose a random frequency division multiplexing (RFDM) method for multicarrier modulation in mobile time-varying channels. Inspired by compressed sensing (CS) technology which use a sensing matrix (with far fewer rows than columns) to sample and compress the original sparse signal simultaneously, while there are many reconstruction algorithms that can recover the original high-dimensional signal from a small number of measurements at the receiver. The approach choose the classic sensing matrix of CS–Gaussian random matrix to compress the signal. However, the signal is not sparse which makes the reconstruction algorithms ineffective. We take full account of the great power of deep neural networks (DNN) to detect the signal as it is an underdetermined equation. The proposed RFDM establishes a novel signal modulation and detection method to target better transmission efficiency, and the simulation results show that the proposed method can achieve good BER, offering a new research paradigm to improve the spectrum efficiency of a multi-subcarrier, multi-antenna, multi-user system.

A Multi-Mode Recognition Method for Broadband Oscillation Based on Compressed Sensing and EEMD

In power systems, the application of wind power generation equipment and power electronic devices leads to an increased frequency of broadband oscillation events, and the detection of oscillation information becomes extremely difficult, due to the limitations of communication bandwidth and the sampling theorem. To ensure the safety and stability of a power system, this paper presents a new recognition method of broadband oscillation information, which combines compressed sensing (CS) technology and an ensemble empirical mode decomposition (EEMD) algorithm to solve the problem of wideband oscillation recognition. Firstly, the broadband oscillation signal data collected by the phasor measuring unit (PMU) is compressed and sampled by a Gaussian random matrix in the substation, then the low-dimensional data obtained is uploaded to the main station. Secondly, in the main station, the subspace pursuit (SP) algorithm is used to reconstruct the low-dimensional signal; the broadband oscillation signal is recovered without losing the main features of the signal. Finally, we use the EEMD algorithm to decompose the reconstructed signal; the intrinsic mode function (IMF) components containing wideband oscillation information are screened by the energy coefficient, and the wideband oscillation information is identified.

Structured orthogonal random features based on DCT for kernel approximation

Random Fourier features are a popular technique used to improve nonlinear kernel methods in large-scale problems. Recent studies have shown that replacing random Gaussian matrices of random feature maps with appropriately scaled random orthogonal matrices, such as SORF, can significantly improve kernel approximation performance. However, the SORF method theoretically requires zero-padding for datasets whose input feature dimension does not meet d=2p for some p∈N+, resulting in increased memory and computation time. To address this limitation, we propose a new structured orthogonal random features method based on discrete cosine transform (SORF-DCT) to approximate Gaussian kernel functions. The SORF-DCT method does not require the input feature dimension to be d=2p. The key to SORF-DCT is that we combine the DCT matrix with diagonal ”sign-flipping” matrices and scale them appropriately using a diagonal matrix whose diagonal elements follow the chi distribution, resulting in a structured orthogonal matrix whose elements follow the standard Gaussian distribution instead of a random Gaussian matrix. SORF-DCT generates D-dimensional structured orthogonal random features in O(Dlogd) time and O(D) memory by employing fast discrete cosine transform, where d and D represent the input feature dimension and expansion dimension, respectively. We prove that SORF-DCT is an unbiased estimator of the Gaussian kernel function and analyze the concentration error bounds. Experimental results on eleven benchmark datasets show that SORF-DCT achieves lower kernel approximation errors, requires less time, and has comparable nonlinear learning capabilities.

Deep Network for Image Compressed Sensing Coding Using Local Structural Sampling

Existing image compressed sensing (CS) coding frameworks usually solve an inverse problem based on measurement coding and optimization-based image reconstruction, which still exist the following two challenges: (1) the widely used random sampling matrix, such as the Gaussian Random Matrix (GRM), usually leads to low measurement coding efficiency, and (2) the optimization-based reconstruction methods generally maintain a much higher computational complexity. In this article, we propose a new convolutional neural network based image CS coding framework using local structural sampling (dubbed CSCNet) that includes three functional modules: local structural sampling, measurement coding, and Laplacian pyramid reconstruction. In the proposed framework, instead of GRM, a new local structural sampling matrix is first developed, which is able to enhance the correlation between the measurements through a local perceptual sampling strategy. Besides, the designed local structural sampling matrix can be jointly optimized with the other functional modules during the training process. After sampling, the measurements with high correlations are produced, which are then coded into final bitstreams by the third-party image codec. Last, a Laplacian pyramid reconstruction network is proposed to efficiently recover the target image from the measurement domain to the image domain. Extensive experimental results demonstrate that the proposed scheme outperforms the existing state-of-the-art CS coding methods while maintaining fast computational speed.

Large deviations and phase transitions in spectral linear statistics of Gaussian random matrices

We evaluate, in the large-N limit, the complete probability distribution P(A,m) of the values A of the sum ∑i=1N|λi|m , where λ i ( i=1,2,…,N ) are the eigenvalues of a Gaussian random matrix, and m is a positive real number. Combining the Coulomb gas method with numerical simulations using a matrix variant of the Wang–Landau algorithm, we found that, in the limit of N→∞ , the rate function of P(A,m) exhibits phase transitions of different characters. The phase diagram of the system on the (A, m) plane is surprisingly rich, as it includes three regions: (i) a region with a single-interval support of the optimal spectrum of eigenvalues, (ii) a region emerging for m < 2 where the optimal spectrum splits into two separate intervals, and (iii) a region emerging for m > 2 where the maximum or minimum eigenvalue ‘evaporates’ from the rest of eigenvalues and dominates the statistics of A. The phase transition between regions (i) and (iii) is of second order. Analytical arguments and numerical simulations strongly suggest that the phase transition between regions (i) and (ii) is of (in general) fractional order p=1+1/|m−1| , where 0<m<2 . The transition becomes of infinite order in the special case of m = 1, where we provide a more complete analytical and numerical description. Remarkably, the transition between regions (i) and (ii) for m⩽1 and the transition between regions (i) and (iii) for m > 2 occur at the ground state of the Coulomb gas which corresponds to the Wigner’s semicircular distribution.

Winding number statistics for chiral random matrices: Averaging ratios of parametric determinants in the orthogonal case

We extend our recent study of winding number density statistics in Gaussian random matrix ensembles of the chiral unitary (AIII) and chiral symplectic (CII) classes. Here, we consider the chiral orthogonal (BDI) case which is the mathematically most demanding one. The key observation is that we can map the topological problem on a spectral one, rendering the toolbox of random matrix theory applicable. In particular, we employ a technique that exploits supersymmetry structures without reformulating the problem in superspace.

Deep Incremental Hashing for Semantic Image Retrieval With Concept Drift

Hashing methods are widely used for content-based image retrieval due to their attractive time and space efficiencies. Several dynamic hashing methods have been proposed for image retrieval tasks in non-stationary environments. However, concept drift problems in non-stationary environment are seldomly considered which lead to significant deterioration of performance. Therefore, we propose Deep Incremental Hashing (DIH). For the learning part, similarity-preserving object codes of each newly arriving data chunk are computed using the product of its label matrix and a random Gaussian matrix generated offline. A point-wise loss function is then devised to guide the learning of a deep hash neural network. To retain the learned knowledge of former chunks, a weighting-based method is utilized to combine different hash tables trained at different time steps to form a multi-table hashing system. Experimental results on 13 simulated concept drift environments show that DIH adapts to non-stationary data environments well and yields better retrieval performance than existing dynamic hashing methods.

A novel method to repair missing vibration data in rolling bearing vibration signals based on improved optimized measurement matrix

Missing data, caused by many factors such as equipment short circuits or data cleaning, affect the accuracy of condition monitoring for rotating machinery. To improve the precision of missing data recovery, a compressed sensing-based vibration data repair method is developed. First, based on the Gaussian random matrix, an improved optimized measurement matrix (OMM) is proposed to accurately sample data. Then, a sparse representation of the vibration signal, through a discrete cosine transform, is utilized to make the noisy vibration signal sparse. Finally, the orthogonal matching pursuit algorithm is employed to reconstruct the missing signal. The effectiveness of the proposed method is verified by analyzing constant and variable speed time series of rolling bearings. Compared with other data repair methods, it is shown that the OMM has a higher repair precision at different loss rates.

Smart meter data transmission based on compressed sensing

To achieve high-precision load recognition for residential electricity consumption and to solve the transmission problem of collecting large amounts of current data from household appliances by smart meters, a data transmission method based on compressed sensing is proposed to reduce data transmission costs. Firstly, a dictionary learning algorithm is used to sparsely represent the current data of household appliances, and the over-complete dictionary matrix is used as the sparse matrix; then a random Gaussian matrix is used as the measurement matrix for data transmission, and finally, regular orthogonal matching pursuit algorithm is used to complete the data reconstruction. The experimental results show that the reconstruction effect of this method is better than the traditional reconstruction method under the same compression ratio.

Design of artificial intelligence image encryption algorithm based on hyperchaos

The artificial intelligence is an inevitable breakthrough direction, and the privacy protection of images in the artificial intelligence environment has become a hotspot. An image encryption algorithm that can be applied in the AI-based work combining compression sensing with Rabinovich hyperchaos is proposed. Firstly, the plaintext image will be transformed into a sparse form by DWT transform. With the improved hyperchaotic sequence, the double reset mechanism of the bit plane and the encryption engine function are designed to perform circular diffusion operation. Finally, the combined random Gaussian measurement matrix will perform compressed sensing sampling on the encrypted image. Simulation experiments show that the encryption algorithm can pass the plaintext sensitivity test (with the NPCR of 0.9978 and the UACI of 0.3337), the correlation test (with the best result of −0.0021), and the information entropy test (with the result of 7.9987), which shows high security to resist several types of attacks.

Joint optimization methods for Gaussian random measurement matrix based on column coherence in compressed sensing

In compressed sensing, a measurement matrix Φ having low coherence with sparse dictionary Ψ can achieve better signal reconstruction performance. To improve the signal reconstruction performance, this paper proposes two joint optimization algorithms for the Gaussian random measurement matrix to minimize the coherence between the measurement matrix Φ and the sparse dictionary Ψ. First, a joint optimization algorithm is proposed that can simultaneously reduce the average mutual coherence μg and the mutual coherence μ based on an alternating projection strategy. Then, to further decrease the coherence between Φ and Ψ, an improved shrinkage method based on K-order cumulative coherence μK is proposed. Furthermore, another joint optimization algorithm is proposed by fusing this improved shrinkage method, which can simultaneously decrease the average mutual coherence μg and the K-order cumulative coherence μK. Simulation results show that the two proposed joint optimization algorithms outperform existing algorithms in reducing coherence and improving reconstruction performance.

The smoothed complexity of Frank–Wolfe methods via conditioning of random matrices and polytopes

Frank–Wolfe methods are popular for optimization over a polytope. One of the reasons is because they do not need projection onto the polytope but only linear optimization over it. To understand its complexity, a fruitful approach in many works has been the use of condition measures of polytopes. Lacoste-Julien and Jaggi introduced a condition number for polytopes and showed linear convergence for several variations of the method. The actual running time can still be exponential in the worst case (when the condition number is exponential). We study the smoothed complexity of the condition number, namely the condition number of small random perturbations of the input polytope and show that it is polynomial for any simplex and exponential for general polytopes. Our results also apply to other condition measures of polytopes that have been proposed for the analysis of Frank–Wolfe methods: vertex-facet distance (Beck and Shtern) and facial distance (Peña and Rodríguez). Our argument for polytopes is a refinement of an argument that we develop to study the conditioning of random matrices. The basic argument shows that for c&gt;1 a d -by- n random Gaussian matrix with n \geq cd has a d -by- d submatrix with minimum singular value that is exponentially small with high probability. This also has consequences on known results about the robust uniqueness of tensor decompositions, the complexity of the simplex method and the diameter of polytopes.

The power quality detection and synchrophasor measurement based on compressive sensing

In order to improve the effect of power quality detection and synchrophasor measurement, this paper studies power quality detection and synchrophasor measurement with the support of compressive sensing technology. Based on the DFT synchrophasor measurement, this paper uses the RDFT recursive formula to simplify the DFT calculation amount, and uses the least squares estimation method to fit the phase angle sequence to calculate the system frequency. According to the obtained frequency estimation value, re-sampling is carried out to correct the original sampling point, and the frequency secondary calculation is iteratively performed, the detection accuracy of high frequency and phase angle, and the linear measurement of the signal is realized by using the Gaussian random measurement matrix, and the compressed sensing signal is obtained. The experimental data statistics show that the model proposed in this paper for power quality detection and synchrophasor measurement based on compressive sensing has a good effect.

On the Condition Number of the Shifted Real Ginibre Ensemble

We derive an accurate lower tail estimate on the lowest singular value $\sigma_1(X-z)$ of a real Gaussian (Ginibre) random matrix $X$ shifted by a complex parameter $z$. Such shift effectively changes the upper tail behaviour of the condition number $\kappa(X-z)$ from the slower $\mathbf{P}(\kappa(X-z)\ge t)\lesssim 1/t$ decay typical for real Ginibre matrices to the faster $1/t^2$ decay seen for complex Ginibre matrices as long as $z$ is away from the real axis. This sharpens and resolves a recent conjecture in [arXiv:2005.08930] on the regularizing effect of the real Ginibre ensemble with a genuinely complex shift. As a consequence we obtain an improved upper bound on the eigenvalue condition numbers (known also as the eigenvector overlaps) for real Ginibre matrices. The main technical tool is a rigorous supersymmetric analysis from our earlier work [arXiv:1908.01653].

A Gaussian High-Dimensional Random Matrix-Based Method for Detecting Abnormal Student Behaviour in Chinese Language Classrooms

In this paper, a Gaussian high-dimensional random matrix approach is used to conduct an in-depth study and analysis of the detection of abnormal student behaviour in Chinese language classrooms, and a corresponding model is designed for practical application. The Gaussian high-dimensional random matrix technique and recurrent neural network technique are applied to build a basic technical framework for intelligent proctoring. An innovative method of human posture estimation is used to complete the proctoring task, and the structure of the Gaussian high-dimensional random matrix method is simplified and the model is compressed to achieve real-time and parallelism of the method. A PolSAR sparse representation classification (OGRM_SRC) algorithm based on orthogonal Gaussian random matrices (OGRM) is proposed to construct an observation dictionary based on polarisation features and OGRM to select typical ground target samples in polarized feature images. The system not only functions as a psychometric assessment of students but also is capable of detecting and analysing abnormal student behaviour based on the improved forest algorithm. The residuals of the pixel to be classified are calculated relative to each atom in the observation dictionary, and the minimum reconstruction residual is used as the classification criterion. The OGRM_SRC algorithm is applied to classify the pixel, and the final classification results are obtained and evaluated for accuracy. We propose a fine-grained action recognition optimisation method for recurrent neural networks, fusing temporal attention information and spatial attention information to improve the action recognition method and improve the model’s ability to recognise fine-grained abnormal behaviours such as looking left, looking right, and copying with head down. Based on this, we conducted some comparative experiments to validate the effectiveness of our work and verified that the system achieved a recognition efficiency of 76.8 FPS and a recognition accuracy of 98.5% in a graphical computing processor.

Sensitivity of (multi)fractal eigenstates to a perturbation of the Hamiltonian

We study the response of an isolated quantum system governed by the Hamiltonian drawn from the Gaussian Rosenzweig-Porter random matrix ensemble to a perturbation controlled by a small parameter. We focus on the density of states, local density of states and the eigenfunction amplitude overlap correlation functions which are calculated exactly using the mapping to the supersymmetric nonlinear sigma model. We show that the susceptibility of eigenfunction fidelity to the parameter of perturbation can be expressed in terms of these correlation functions and is strongly peaked at the localization transition: It is independent of the effective disorder strength in the ergodic phase, grows exponentially with increasing disorder in the fractal phase and decreases exponentially in the localized phase. As a function of the matrix size, the fidelity susceptibility remains constant in the ergodic phase and increases in the fractal and in the localized phases at modestly strong disorder. We show that there is a critical disorder strength inside the insulating phase such that for disorder stronger than the critical the fidelity susceptibility decreases with increasing the system size. The overall behavior is very similar to the one observed numerically in a recent work by Sels and Polkovnikov [Phys. Rev. E 104, 054105 (2021)] for the normalized fidelity susceptibility in a disordered XXZ spin chain.

Optimal orthogonal group synchronization and rotation group synchronization

Abstract We study the statistical estimation problem of orthogonal group synchronization and rotation group synchronization. The model is $Y_{ij} = Z_i^* Z_j^{*T} + \sigma W_{ij}\in{\mathbb{R}}^{d\times d}$ where $W_{ij}$ is a Gaussian random matrix and $Z_i^*$ is either an orthogonal matrix or a rotation matrix, and each $Y_{ij}$ is observed independently with probability $p$. We analyze an iterative polar decomposition algorithm for the estimation of $Z^*$ and show it has an error of $(1+o(1))\frac{\sigma ^2 d(d-1)}{2np}$ when initialized by spectral methods. A matching minimax lower bound is further established that leads to the optimality of the proposed algorithm as it achieves the exact minimax risk.

Many body density of states in the edge of the spectrum: non-interacting limit

In noninteracting limit, the density of states (dos) of a many body system can be expressed as a convolution of the single body dos of its subunits. We use the formulation to derive, in the edge of the spectrum, a differential equation for the ensemble averaged many body dos that is relatively easier to solve. Our analysis, based on the systems in which the subunits can be modelled by a Gaussian or Wishart random matrix ensemble, indicates that a rescaling of energy by the number of subunits leaves the many body dos in a mathematically invariant form.

Detecting few-body quantum chaos: out-of-time ordered correlators at saturation

We study numerically and analytically the time dependence and saturation of out-of-time ordered correlators (OTOC) in chaotic few-body quantum-mechanical systems: quantum Henon-Heiles system (weakly chaotic), BMN matrix quantum mechanics (strongly chaotic) and Gaussian random matrix ensembles. The growth pattern of quantum-mechanical OTOC is complex and nonuniversal, with no clear exponential regime at relevant timescales in any of the examples studied (which is not in contradiction to the exponential growth found in the literature for many-body systems, i.e. fields). On the other hand, the plateau (saturated) value of OTOC reached at long times decreases with temperature in a simple and universal way: exp(const./T2) for strong chaos (including random matrices) and exp(const./T) for weak chaos. For small matrices and sufficiently complex operators, there is also another, high-temperature regime where the saturated OTOC grows with temperature. Therefore, the plateau OTOC value is a meaningful indicator of few-body quantum chaos. We also discuss some general consequences of our findings for the AdS/CFT duality.

Generalized residual ratio thresholding

Recovering sparse high dimensional vectors from low dimensional linear measurements is an important compressive sensing (CS) problem. Many CS algorithms assume a priori knowledge of parameters like signal sparsity or noise variance which are unavailable in most real-life problems. It is also difficult to efficiently estimate these parameters with finite sample guarantees. This article proposes a support recovery technique called generalized residual ratio thresholding (GRRT) that can operate many popular sparsity or noise variance dependent sparse recovery algorithms in a sparsity and noise variance oblivious fashion with finite sample guarantees when the noise is composed of independent and identically distributed (i.i.d) Gaussian random variables and design matrix satisfies certain regularity conditions. Numerical simulations and theoretical results in sparse estimation scenarios like single measurement vector, multiple measurement vectors, block sparsity etc. indicate that the performance of algorithms operated using GRRT is comparable to the performance of same algorithms when operated with a priori knowledge of sparsity and noise variance.