Distribution Of Singular Values Research Articles

Random-Matrix Models of Monitored Quantum Circuits

We study the competition between Haar-random unitary dynamics and measurements for unstructured systems of qubits. For projective measurements, we derive various properties of the statistical ensemble of Kraus operators analytically, including the purification time and the distribution of Born probabilities. The latter generalizes the Porter–Thomas distribution for random unitary circuits to the monitored setting and is log-normal at long times. We also consider weak measurements that interpolate between identity quantum channels and projective measurements. In this setting, we derive an exactly solvable Fokker–Planck equation for the joint distribution of singular values of Kraus operators, analogous to the Dorokhov–Mello–Pereyra–Kumar (DMPK) equation modelling disordered quantum wires. We expect that the statistical properties of Kraus operators we have established for these simple systems will serve as a model for the entangling phase of monitored quantum systems more generally.

Dynamic relationship between the XRP price and correlation tensor spectra of transaction networks

The emergence of cryptoassets has sparked a paradigm shift in the world of finance and investment, ushering in a new era of digital assets with profound implications for the future of currency and asset management. A recent study showed that during the bubble period around the year, 2018, the price of cryptoasset, XRP has a strong anti correlation with the largest singular values of the correlation tensors obtained from the weekly XRP transaction networks. In this study, we provide a detailed analysis of the method of correlation tensor spectra for XRP transaction networks. We calculate and compare the distribution of the largest singular values of the correlation tensor using the random matrix theory with the largest singular values of the empirical correlation tensor. We investigate the correlation between the XRP price and the largest singular values for a period spanning two years. We also uncover the distinct dependence between the XRP price and the singular values for bubble and non-bubble periods. The significance of the time evolution of singular values is shown by comparison with the evolution of singular values of the reshuffled correlation tensor. Furthermore, we identify a set of driver nodes in the transaction networks that drives the market during the bubble period using singular vectors.

GLT sequences and automatic computation of the symbol

Spectral and singular value symbols are valuable tools to analyse the eigenvalue or singular value distributions of matrix-sequences in the Weyl sense. More recently, Generalized Locally Toeplitz (GLT) sequences of matrices have been introduced for the spectral/singular value study of the numerical approximations of differential operators in several contexts. As an example, such matrix-sequences stem from the large linear systems approximating Partial Differential Equations (PDEs), Fractional Differential Equations (FDEs), Integro Differential Equations (IDEs), using any discretization on reasonable grids via local methods, such as Finite Differences, Finite Elements, Finite Volumes, Isogeometric Analysis, Discontinuous Galerkin etc. Studying the asymptotic spectral behaviour of GLT sequences is useful in analysing classical techniques for the solution of the corresponding PDEs/FDEs/IDEs and in designing novel fast and efficient methods for the corresponding large linear systems or related large eigenvalue problems. The theory of GLT sequences, in combination with the results concerning the asymptotic spectral distribution of perturbed sequences of matrices, is one of the most powerful and successful tools for computing the spectral symbol f. In this regard, it would be beneficial to design an automatic procedure to compute the spectral symbols of such matrix-sequences and Ahmed Ratnani partially pursued it. Here, in the case of one-dimensional and two-dimensional differential problems, we continue in this direction by proposing an automatic procedure for computing the symbol of the underlying sequences of matrices, assuming that it is a GLT sequence satisfying mild conditions.

A feature-recombinant asynchronous deep reservoir computing for modeling time series data

Modeling time series data is an important issue in many areas. Reservoir computing (RC) is a promising tool to build time series model due to its dynamic characteristic and simple training way. Asynchronous deep reservoir computing (ADRC) is an improvement version of the traditional RC. It solves time-dependent tasks more efficiently than traditional RC because of its rich dynamics and flexible short-term memory (STM). Nevertheless, it has been an open issue to design RC’s or ADRC’s reservoir topology owing to large amounts of random factors. To promote the solution of this problem, the paper proposes a feature-recombinant ADRC (FR-ADRC) for modeling time series data. In the FR-ADRC scheme, the first sub-reservoir is designed as the feature-adaptive layer, and a trainable matrix C is introduced into this layer. By learning C, the singular value (SV) distribution of the first layer could be adjusted. Further, the principal components of the reservoir topology can be extracted by the principal component analysis (PCA). Then a new temporary reservoir is constructed by recombining these extracted components. The subsequent information processing is carried out based on the recombinant reservoir, which can be adaptive to the input signals. The validity of the FR-ADRC is tested by modeling some numerical and real-life time series data. Experimental results show that the proposed approach is better than the traditional ADRC in modeling precision, generalization ability and stability.

Interval strategy-based regularization approach for force reconstruction with multi-source uncertainties

A force reconstruction method for uncertain structures instead of direct measurement is proposed in this paper. Considering the multi-source uncertainties as unknown-but-bounded (UBB) parameters including unknown structural dynamics and measurement errors, this study proposed a novel force reconstruction method using an interval strategy-based regularization approach with truncated singular value decomposition (TSVD). To overcome the drawbacks of incompleted models and limited information, the overall modelling and inversion of the interval force are reconstructed based on the non-probabilistic uncertainty theory. First, The interval uncertainty-based kernel function for force inversion is deduced using the second-order interval perturbation method, in which the uncertainties of the reconstructed force originate from the uncertain singular values and measurement errors. Based on the regularization of the TSVD, the interval bounds of the singular values can be accurately obtained. Combining the two typical regularization approaches, a novel interval strategy-based regularization approach is proposed to optimize the best regularization parameter based on the uncertain interval distribution of singular values. This novel strategy can not only balance the two typical regularization approaches but also reduce the burden of force inversion. Two numerical examples including a truss and space capsule are applied to access the proposed method, and the estimated accurate force can prove the effectiveness.

Broadband Beamforming Using the Least Square Method Improved via Adaptive Diagonal Loading

The least square method (LSM) is a commonly used beamforming method. However, it limits the beamforming frequency bandwidth and is vulnerable to noise; therefore, diagonal loading is used to overcome these limitations. The diagonal loading proposed in this study can be used for broadband beamforming in cases in which noise does not exist, and it makes the loading factor adaptive to a singular value distribution. The simulation results show that the proposed method works stably above the beamforming frequency limit, even for an extended beamforming frequency. We expect that the proposed method can be used to form frequency-invariant beam patterns used in monopulse algorithms to detect signals of unknown frequencies.

Interpreting Deep Graph Convolutional Networks with Spectrum Perspective

Graph convolutional network (GCN) architecture is the basis of many neural networks and has been widely used in processing graph-structured data. When dealing with large and sparse data, deeper GCN models are often required. However, the models suffer from performance degradation as the number of layers increases. The mainstream attribution of the current research is over-smoothing, and there are also gradient vanishing, training difficulties, etc., so a consensus cannot be reached. In this paper, we theoretically analyze the degradation problem by adopting spectral graph theory to globally consider the propagation and transformation components of the GCN architecture, and conclude that the over-smoothing problem caused by the propagation matrices is not the key factor for performance degradation. Afterwards, in addition to using conventional experimental methods, we proposed an experimental analysis strategy under the guidance of random matrix theory to analyze the singular value distribution of the model weight matrix. We concluded that the key factor leading to the degradation of model performance is the transformation component. In the context of a lack of consensus on the problem of model performance degradation, the paper proposes a systematic analysis strategy, as well as theoretical and empirical evidence.

Neutrino mass square ratio and neutrinoless double-beta decay in random neutrino mass matrices

AbstractWe study neutrino mass anarchy in the Dirac neutrino, seesaw, and double-seesaw models. Assuming the anarchy hypothesis, the mass matrices are random and distributed in accordance with the Gaussian measure. We focus on the distributions of the mass square ratio of the light neutrinos and examine which of these models shows a peak in the probability distribution around the experimental value. We show that the peak position depends on the number of random matrix products. We find that the light neutrino mass hierarchy becomes larger as the number of random matrix products is increased and the seesaw model with the random Dirac and Majorana mass matrices is the most likely to realize the current experimental data. We also investigate the distributions of the effective Majorana mass for neutrinoless double-beta decay. We find that the effective Majorana mass is smaller than the experimental upper bound and tends to be smaller as the number of random matrix products increases because the light neutrino masses become more hierarchical. We argue that the tendency for lighter neutrino masses to become more hierarchical as the number of products in the random matrix increases can be understood from the probability distribution of singular values in random matrix theory.

Singular value distribution of dense random matrices with block Markovian dependence

A block Markov chain is a Markov chain whose state space can be partitioned into a finite number of clusters such that the transition probabilities only depend on the clusters. Block Markov chains thus serve as a model for Markov chains with communities. This paper establishes limiting laws for the singular value distributions of the empirical transition matrix and empirical frequency matrix associated to a sample path of the block Markov chain whenever the length of the sample path is Θ(n2) with n the size of the state space.The proof approach is split into two parts. First, we introduce a class of symmetric random matrices with dependent entries called approximately uncorrelated random matrices with variance profile. We establish their limiting eigenvalue distributions by means of the moment method. Second, we develop a coupling argument to show that this general-purpose result applies to the singular value distributions associated with the block Markov chain.

A note on the spectral analysis of matrix sequences via GLT momentary symbols: from all-at-once solution of parabolic problems to distributed fractional order matrices

The first focus of this paper is the characterization of the spectrum and the singular values of the coefficient matrix stemming from the discretization of a parabolic diffusion problem using a space-time grid and secondly from the approximation of distributed-order fractional equations. For this purpose we use the classical GLT theory and the new concept of GLT momentary symbols. The first permits us to describe the singular value or eigenvalue asymptotic distribution of the sequence of the coefficient matrices. The latter permits us to derive a function that describes the singular value or eigenvalue distribution of the matrix of the sequence, even for small matrix sizes, but under given assumptions. The paper is concluded with a list of open problems, including the use of our machinery in the study of iteration matrices, especially those concerning multigrid-type techniques.

A WTC suppression algorithm for weather radar based on weighted schatten p-norm minimization

ABSTRACT The Matrix Completion (MC) method can effectively suppress the wind turbine clutter (WTC) of the weather radar, however its recovery accuracy suffers from the low signal-to-noise ratio (SNR) of the real measured weather signal. Moreover, the singular value distribution of the observation matrix constructed by the measured radar data has no sharp cut-off characteristic, which results the rank can not be well approximated and also reduces the recovery accuracy. In this paper, a non-convex MC algorithm based on weighted Schatten p-norm is proposed for the WTC suppression in Doppler frequency domain. Experiments are carried out on the measured signal data, and the results demonstrate that compared with the commonly used nuclear norm minimization (NNM) model, the weighted Schatten p-norm minimization (WSNM) model can further improve the recovery accuracy of weather signal and realizes the superior estimation performance of spectral moment.

Random matrix analysis of deep neural network weight matrices.

Neural networks have been used successfully in a variety of fields, which has led to a great deal of interest in developing a theoretical understanding of how they store the information needed to perform a particular task. We study the weight matrices of trained deep neural networks using methods from random matrix theory (RMT) and show that the statistics of most of the singular values follow universal RMT predictions. This suggests that they are random and do not contain system specific information, which we investigate further by comparing the statistics of eigenvector entries to the universal Porter-Thomas distribution. We find that for most eigenvectors the hypothesis of randomness cannot be rejected, and that only eigenvectors belonging to the largest singular values deviate from the RMT prediction, indicating that they may encode learned information. In addition, a comparison with RMT predictions also allows to distinguish networks trained in different learning regimes-from lazy to rich learning. We analyze the spectral distribution of the large singular values using the Hill estimator and find that the distribution cannot in general be characterized by a tail index, i.e., is not of power-law type.

Tensor denoising of multidimensional MRI data.

To develop a denoising strategy leveraging redundancy in high-dimensional data. The SNR fundamentally limits the information accessible by MRI. This limitation has been addressed by a host of denoising techniques, recently including the so-called MPPCA: principal component analysis of the signal followed by automated rank estimation, exploiting the Marchenko-Pastur distribution of noise singular values. Operating on matrices comprised of data patches, this popular approach objectively identifies noise components and, ideally, allows noise to be removed without introducing artifacts such as image blurring, or nonlocal averaging. The MPPCA rank estimation, however, relies on a large number of noise singular values relative to the number of signal components to avoid such ill effects. This condition is unlikely to be met when data patches and therefore matrices are small, for example due to spatially varying noise. Here, we introduce tensor MPPCA (tMPPCA) for the purpose of denoising multidimensional data, such as from multicontrast acquisitions. Rather than combining dimensions in matrices, tMPPCA uses each dimension of the multidimensional data's inherent tensor-structure to better characterize noise, and to recursively estimate signal components. Relative to matrix-based MPPCA, tMPPCA requires no additional assumptions, and comparing the two in a numerical phantom and a multi-TE diffusion MRI data set, tMPPCA dramatically improves denoising performance. This is particularly true for small data patches, suggesting that tMPPCA can be especially beneficial in such cases. The MPPCA denoising technique can be extended to high-dimensional data with improved performance for smaller patch sizes.

Using random matrix theory to quantify pulmonary fibrosis: Investigating the effect of time window duration

Random matrix theory (RMT) exploits the distribution of singular values of the inter-element response matrix (IRM). If multiple scattering dominates the propagation, the singular value distribution follows the quarter circle law. However, dominance of single scattering results in Henkel function behavior of the singular value distribution. In our previous work, we have shown that this can be exploited to estimate severity of bleomycin-induced fibrosis (measured by histology) in rodent lungs. We showed that E(x), the expected value of the singular value distribution, as well as the singular value with the highest probability, correlated significantly with histology scores. Here, we investigate the sensitivity of these metrics to the time window duration used for the Singular Value Decomposition of the IRM, which is performed in the frequency domain, using overlapping time windows. A linear transducer with a central frequency of 7.8 MHz and a Verasonics scanner were used to obtain IRMs in 24 rat lungs. Different degrees of pulmonary fibrosis were induced using bleomycin in 18 rats while 6 rats were left as controls. The IRMs were time-windowed with different duration of 2T, 4T, and 6T where T is the transmitted pulse period. E(x) and were evaluated for each time window duration for all rat lungs. For all time window durations significant correlations were observed between and E(x), and histology scores. Wilcoxon ranksum tests show that the distributions obtained for and E(x) are not affected by the time window duration.

Bearing fault diagnosis based on singular value distribution of impulse response segment

Extracting periodic impact features from vibration signals has always been a key issue in the fault diagnosis of rolling element bearings. However, the repetitive impacts induced by localized defects are difficult to identify due to the presence of background noise and interferences. A novel approach for bearing fault diagnosis based on singular value distribution of impulse response segment is proposed. The characteristics of singular value decomposition (SVD) of the impulse response are analyzed, and the relationship between the matrix row number and the bandwidth of subspace is estimated quantitatively. According to the unique distribution of singular values, the double-order attenuation ratio (DAR) is designed to evaluate the transient component of the short-time segment. Then, by segmenting the vibration signal, the time-dependent DAR sequences are obtained, which can be used to locate the impacts in the signal. Eventually, the fault-related cyclo-stationarity in DAR sequences is enhanced by autocorrelation and measured by the Gini index. The effectiveness of the proposed method is verified by simulation and bearing fault datasets, in contrast to the state-of-art algorithms.

Eigenvalue distribution of large random matrices arising in deep neural networks: Orthogonal case

This paper deals with the distribution of singular values of the input–output Jacobian of deep untrained neural networks in the limit of their infinite width. The Jacobian is the product of random matrices where the independent weight matrices alternate with diagonal matrices whose entries depend on the corresponding column of the nearest neighbor weight matrix. The problem has been considered in the several recent studies of the field for the Gaussian weights and biases and also for the weights that are Haar distributed orthogonal matrices and Gaussian biases. Based on a free probability argument, it was claimed in those papers that, in the limit of infinite width (matrix size), the singular value distribution of the Jacobian coincides with that of the analog of the Jacobian with special random but weight independent diagonal matrices, the case well known in random matrix theory. In this paper, we justify the claim for random Haar distributed weight matrices and Gaussian biases. This, in particular, justifies the validity of the mean field approximation in the infinite width limit for the deep untrained neural networks and extends the macroscopic universality of random matrix theory to this new class of random matrices.

Weighted p-radial distributions on Euclidean and matrix p-balls with applications to large deviations

A probabilistic representation for a class of weighted p-radial distributions, based on mixtures of a weighted cone probability measure and a weighted uniform distribution on the Euclidean ℓpn-ball, is derived. Large deviation principles for the empirical measure of the coordinates of random vectors on the ℓpn-ball with distribution from this weighted measure class are discussed. The class of p-radial distributions is extended to p-balls in classical matrix spaces, both for self-adjoint and non-self-adjoint matrices. The eigenvalue distribution of a self-adjoint random matrix, chosen in the matrix p-ball according to such a distribution, is determined. Similarly, the singular value distribution is identified in the non-self-adjoint case. Again, large deviation principles for the empirical spectral measures for the eigenvalues and the singular values are presented as an application.

Quantifying severity of lung fibrosis in rodents using random matrix theory

We investigate random matrix theory (RMT) as a tool to detect and quantify pulmonary fibrosis in rodents in vivo. Highly scattering structures such as lung alveoli result in specific characteristics in the distribution of singular values of the inter-element response matrix (IRM). When multiple scattering dominates (healthy lung), the distribution of singular values of the IRM is expected to follow a quarter circle law. However, when single scattering regime dominates, the singular values distribution is closer to a Henkel distribution. We propose to exploit this feature to detect pulmonary fibrosis and quantify its severity. Two metrics are defined to describe the singular value distribution: the expected value E(x), which is the weighted average of all singular values, and, the singular value with the highest probability. A 128-element linear transducer operating at 7.8 MHz and a Verasonics scanner were used to collect IRMs from 6 normal and 18 rat lungs with bleomycin-induced fibrosis in vivo. Significant correlations were observed between E(x) (r = −0.46, p = 0.02) and (r = 0.52, p = 0.01) with the severity of fibrosis independently assessed by histology. These preliminary results show the potential of RMT metrics E(x) and to quantify structural changes in the lung parenchyma.

Local Law for Singular Values of Oscillatory Matrices

AbstractWe continue the study of spectra of oscillatory random matrices with fully dependent rows. Motivated by the $d$-dimensional skew-shift dynamics from ergodic theory, we introduce the $N\times N$ random matrices $$ \begin{align*} &X_{j,k}=\exp\left(2\pi \textrm{i} \sum_{q=1}^d\ \omega_{j,q} k^q\right), \end{align*}$$where $\{\omega _{j,q}\}_{1\leq j\leq N, 1\leq q\leq d}$ is a collection of i.i.d. random variables and $d$ is a fixed integer. We prove that as $N\to \infty $ the distribution of singular values converges to the local Marchenko–Pastur law up to scales $N^{-\theta _d}$ for an explicit $\theta _d>0$, as long as $d\geq 18$. Our approach provides a novel mechanism—deterministic oscillatory cancellations—for universal spectral laws. The proof blends techniques from random matrix theory, harmonic analysis, and analytic number theory such as strong estimates on the number of solutions to Diophantine equations in the form of Vinogradov’s main conjecture, proved by Bourgain–Demeter–Guth.

A singular value decomposition entropy approach to instability analysis in BWR’s

The power reactor is routinely measured and has been widely used for characterization of the reactor core dynamics. This signal exhibits complex dynamics, including a combination of large- and small-amplitude fluctuations. This work proposed a method based on singular value decomposition (SVD) to quantify the complexity of the power signal. The underlying idea is to assess the signal complexity in terms of the entropy computed from the singular value distribution and compare such entropy with that obtained from a random process. In this way, a distance to randomness was introduced, which was used as an index for monitoring the signal dynamics. The methodology was used to study benchmark power instability events of Laguna Verde, Forsmark and Ringhals NPPs. The analysis was carried out for a wide range of time scales and showed that instabilities are associated to marked reductions of the SVD entropy, and hence to large distances from randomness. The results were contrasted with a traditional statistical method based on the autocovariance function, and the results showed that the SVD entropy approach was more sensible to stability variations. The SVD entropy showed sufficient sensitivity to detect short-lasted unstable events appearing in in-phase and out-of-phase power oscillations.