A Note on Some Extensions of the Matrix Angular Central Gaussian Distribution

Sparse representation of local spatial-temporal features with dimensionality reduction for motion recognition

The structure of this survey is the following: first we repeat the first, now 20 years old, proof of the Strong Circular Law for non Hermitian random matrices under the assumptions that the probability densities of the random entries exist and they satisfy the Lyapunov condition. (see [4] and the sketch of the proof of this law in the paper V-transform , Dopovidi Akademii nauk Ukrainskoi RSR, Seria A: Fizyko-Matematychni ta technichni nauky, 1982, N3, pp.5-6.(see[2])). Then we prove the Strong V-Law for random matrices of general form, i.e. the entries of matrices have nonzero expectations and different variances. In this case the V-Law means that the support of the accompanying spectral density of eigenvalues looks like several water or mercury drops on the table. According to the distances between the centers of these drops, these drops might not touch each other or can merge making fanciful shapes represented at the end of the paper, Figures 13–16. If the distances between the centers of these drops are large enough, we have several separate almost circular drops. In Part II we prove the weak Circular law, (where we do not require the above assumptions on the densities and Lyapunov condition) but instead of these two conditions we require that the Weak Circular Law condition is fulfilled: for some In Part III of the survey we give the formulation of the Circular Law for Unitary random matrices from the so called Class N 12 of random Unitary matrices(see[20]). In this case the Circular law means that the support of the accompanying spectral density of eigenvalues looks like several drops of water, oil or mercury on the table and inside of some drops it is possible that some dry circles appear. If the distances between the centers of these drops are large enough we have several almost circular drops as for corresponding description of limit spectral density for the Hermitian random matrices. In Part II we prove the following Weak Global Circular Law( Weak V -Law) which generalizes the Strong Global Circular Law (Strong V -Law) : For every n, let the random entries of the complex matrix be independent, the Weak Circular Law condition is fulfilled and where and are square complex nonrandom matrices, det A n ≠ 0, det B n ≠ 0. Moreover we require the so-called Central Limit V -condition and Border V -condition are fulfilled. See the precise formulation of these condition in the Section 1 of this paper and in the Part II. Roughly speaking, the Central Limit V -condition means that we can apply the central limit theorem for the heights of random parallelogram (obtained from the random matrix). The Border V -condition means that we can exclude from V -transform some small region between two regions were we apply two different method of limit theorems. Then, in probability, for almost all x and y where λ k are eigenvalues of the matrix A n Ξ n B n + C n ; the Global probability density p n,α ( t , s ) = ( ∂ 2 / ∂t ∂s ) F n,α ( t, s ) is equal to where the analytic function m n ( y; t; s ) satisfies the canonical equation K 26 (see[12, 13, 20]) In particular, we have gathered in this paper three proofs of the STRONG CIRCULAR LAW: 1. the first proof based on the V -regularization of the determinant of random matrix and on the application of an inequality of the Berry-Esseen type[4]. 2. the second proof based on the central limit theorem for random determinants and on the replacement a square matrix by rectangular random matrix[11]. 3. the third proof based on the regularization of random determinant and replacement square matrix by rectangular matrix[18]. Therefore, we have proved triply the Circular law: Let be a complex random matrix whose entries are defined on a common probability space, are independent for any n = 1, 2, ... and are such that and . Further, assume that there exist either densities of the real parts of the random entries (or densities of the imaginary parts of the random entries and, for some δ > 0) and β > 1 Then, with probability one, where is a normalized spectral function and λ k are eigenvalues of the matrix

The structure of this survey is the following: at first we repeat in Part I the first 20 years old proof of the strong Elliptic law for random matrices Then in Part II we prove the strong Elliptic law for random matrices Ξ n of the general form, i.e. when their diagonal entries have nonzero expectations, and when we do not require the existence of the probability densities of the entries of random matrices, but instead of Lyapunov condition we require the boundedness of the absolute moments of the entries of the order 4 + δ , δ &gt; 0. In this case the Elliptic law means that the support of the accompanying spectral density of eigenvalues looks like the picture of several galaxies made by telescope. If the distances between the centers of these galaxies are large enough we have several almost elliptical galaxies. We also present the sand clock law density for the sum of diagonal complex matrix and the symmetric random matrix belonging the domain of attraction of the Semicircular law. After this we prove in the Part III the weak Elliptic law, when all entries can have nonzero expectations. At the end of survey we give the proof of the Elliptic law for the Unitary random matrices from the so called Class N 12 of random Unitary matrices and give the over turned stool law density for the sum of diagonal complex matrix A n and where Ξ n belongs to the domain of attraction of the Circular law. We also establish in Part III for Elliptic Law at the first time the convergency rate and the Central Limit Theorem. These statements are based on the VICTORIA -transform of random matrix which is the abbreviation of the following words: Very Important Computational Transformation Of Random Independent Arrays. We follow the main strategy of the theory of limit theorems of the probability theory, i.e. we try to solve the problem of description of all limits of normalized spectral functions where λ k ( A n Ξ n B n + C n ) are eigenvalues of the matrix A n Ξ n B n + C n , A n , B n , and C n are nonrandom matrices, under general (as only possible) conditions on the entries of random matrices Ξ n , χ is the indicator function. We emphasize that the spectral theory of Hermitian random matrices is rather profound. For example, in 1975 in [3] V. Girko proved the general stochastic canonical equation for ACE (Asymptotically Constant Entries)-symmetric matrices: Assume that for any n , the random entries of a symmetric matrix are independent and they are asymptotically constant entries (ACE), i.e., for any ε &gt; 0, and τ &gt; 0 is an arbitrary constant, and that, for every 0 ≤ u ≤ 1 and 0 ≤ v ≤ 1, where the symbol denotes the weak convergence of distribution when n → ∞, is a nondecreasing function with bounded variation in z and continuous in u and v in the domain 0 ≤ u , v ≤ 1. Then, with probability one, for almost all x , where λ k (Ξ n × n ) are eigenvalues, F ( x ) is a distribution function whose Stieltjes transform satisfies the relation G α ( x , y , t ), as a function of x , is a distribution function satisfying the regularized stochastic canonical equation K 3 [3, 23] at the points x of continuity, is a random real functional whose Laplace transform of one-dimensional distribution is equal to The integrand <jats:inli

This paper proposes a novel acoustic model transformation method for speech recognition based on random projections. Random projections have been suggested as a means of dimensionality reduction, where the original data are projected onto a subspace using a random matrix. Moreover, as we are able to produce various random matrices, it may be possible to find a transform matrix that is superior to conventional transformation matrices among random matrices. In our previous work, a random-projection-based feature combination technique has been proposed but had a high computational cost. In order to deal with this cost, in this paper, we introduce random projections on the acoustic model domain, where linear transformations are applied to an acoustic model using random matrices. Its effectiveness is confirmed by word recognition experiments on noisy speech.

Elliptic Law

In this paper the Bhattacharyya distance and the divergence are derived as two different measures of target class separability based on the central complex multivariate Gaussian and Wishart distributions with unequal covariance matrices. The derived Bhattacharyya distances for the two distributions, respectively, differ only in terms of a simple multiplier, i.e. the number of degrees of freedom (also known as number of looks in polarimetric synthetic aperture radar data). The same aspect was observed for the divergence. Furthermore, the Bhattacharyya distance was found to be proportional to the Bartlett distance, while the divergence is proportional to the symmetrized normalized log-likelihood distance. The use of the Bhattacharyya distance and the divergence as separability measures of target classes was demonstrated by using NASA/JPL AIRSAR POLSAR data and was benchmarked against the Euclidean distance. From the results, both the Bhattacharyya distance and the divergence were found to perform consistently in measuring the separability of target classes.

The Gaussian distribution is commonly used to model uncertainty for all kind of problems. However, for directional data like fiber orientations in injection molding simulations the canonical choice is the so called angular central Gaussian (ACG) distribution, which arises as analytical solution to Jeffrey's equation which is used to model the orientation of a elliptical fiber suspended in a flow field. Computations are favorably performed using moments of the density instead of the density itself, leading to the so called Folgar‐Tucker equation. In this differential equation for the second order moment also the fourth order moment arises, which has to be expressed in terms of the second order moment in order to close the equation. This is called the “closure problem”, which has been addressed in many publications with various proposals for the solution, among which the fourth order moment of the ACG distribution represents the exact solution. The ACG is obtained by normalizing a centered multivariate Gaussian and therefore the individual components of the ACG distribution are no longer independent and the moments of the ACG distribution are coupled with its covariance parameter in a more complicated way in the form of elliptic integrals. This beautiful distribution is not very well studied and hardly used for directional statistics, where more empirical or wrapped distributions are employed instead.In this article we compare the ACG to the Bingham distribution and discuss the current status of computing moments as well as the analytical and numerical solution of the closure problem.

Gaussian distributions on Riemannian symmetric spaces of nonpositive curvature

Consider a $N\times n$ matrix $\Sigma_n=\frac{1}{\sqrt{n}}R_n^{1/2}X_n$, where $R_n$ is a nonnegative definite Hermitian matrix and $X_n$ is a random matrix with i.i.d. real or complex standardized entries. The fluctuations of the linear statistics of the eigenvalues \[\operatorname {Trace}f \bigl(\Sigma_n\Sigma_n^*\bigr)=\sum_{i=1}^Nf(\lambda_i),\qquad (\lambda_i)\ eigenvalues\ of\ \Sigma_n\Sigma_n^*,\] are shown to be Gaussian, in the regime where both dimensions of matrix $\Sigma_n$ go to infinity at the same pace and in the case where $f$ is of class $C^3$, that is, has three continuous derivatives. The main improvements with respect to Bai and Silverstein's CLT [Ann. Probab. 32 (2004) 553-605] are twofold: First, we consider general entries with finite fourth moment, but whose fourth cumulant is nonnull, that is, whose fourth moment may differ from the moment of a (real or complex) Gaussian random variable. As a consequence, extra terms proportional to $ \vert \mathcal{V}\vert ^2=\bigl|\mathbb{E}\bigl(X_{11}^n\bigr) ^2\bigr|^2$ and $\kappa=\mathbb{E}\bigl \vert X_{11}^n\bigr \vert ^4-\vert {\mathcal{V}}\vert ^2-2$ appear in the limiting variance and in the limiting bias, which not only depend on the spectrum of matrix $R_n$ but also on its eigenvectors. Second, we relax the analyticity assumption over $f$ by representing the linear statistics with the help of Helffer-Sj\"{o}strand's formula. The CLT is expressed in terms of vanishing L\'{e}vy-Prohorov distance between the linear statistics' distribution and a Gaussian probability distribution, the mean and the variance of which depend upon $N$ and $n$ and may not converge.

In this paper, we formulate the manipulator Jacobian matrix in a probabilistic framework based on the random matrix theory (RMT). Due to the limited available information on the system fluctuations, the parametric approaches often prove to be inadequate to appropriately characterize the uncertainty. To overcome this difficulty, we develop two RMT-based probabilistic models for the Jacobian matrix to provide systematic frameworks that facilitate the uncertainty quantification in a variety of complex robotic systems. One of the models is built upon direct implementation of the maximum entropy principle that results in a Wishart random perturbation matrix. In the other probabilistic model, the Jacobian matrix is assumed to have a matrix-variate Gaussian distribution with known mean. The covariance matrix of the Gaussian distribution is obtained at every time point by maximizing a Shannon entropy measure (subject to Jacobian norm and covariance positive semidefiniteness constraints). In contrast to random variable/vector based schemes, the benefits of the proposed approach now include: (i) incorporating the kinematic configuration and complexity in the probabilistic formulation; (ii) achieving the uncertainty model using limited available information; (iii) taking into account the working configuration of the robotic systems in characterization of the uncertainty; and (iv) realizing a faster simulation process. A case study of a 2R serial manipulator is presented to highlight the critical aspects of the process.

Linear Video Coding and Transmission (LVCT) schemes, pioneered by the SoftCast architecture, have been recently proposed as an alternative solution for wireless broadcast applications where users may experience severe channel quality fluctuations. LVCT schemes offer a graceful video quality adaptation to the user experienced channel quality. This is made possible by relying on linear operators only (decorrelation transform, power allocation, whitening). The LVCT whitening process aims at reducing the impact of slice loss in presence of fades. It is mainly performed by Hadamard transform which is not adapted to all input sizes. Although alternatives to the Hadamard transform have been proposed in the literature, no performance comparison has been performed. In this paper, we compare the performance of SoftCast using three different whitening matrices, namely, Hadamard, Inverse Discrete Cosine Transform (IDCT) and Random matrices. Three scenarios are considered for the comparison: no packet loss, random packet losses and bursty packet losses. Simulation results confirm that these matrices offer similar reconstructed video quality in a lossless scenario, while whitening using a random orthogonal matrix outperforms whitening using Hadamard or IDCT matrices in all lossy scenarios.

This paper describes a selection technique of an optimum random matrix using a genetic algorithm for speech recognition based on random projections. Random projections have been suggested as a means of dimensionality reduction, where the original data are projected onto a subspace using a random matrix. Moreover, as we are able to produce various random matrices, it may be possible to find a transform matrix that is superior to conventional transformation matrices among random matrices. In this paper, a genetic algorithm is introduced to find an optimum random matrix. Its effectiveness is confirmed by word recognition experiments.

This article proposes a method to consistently estimate functionals $\frac1p\sum_{i=1}^pf(λ_i(C_1C_2))$ of the eigenvalues of the product of two covariance matrices $C_1,C_2\in\mathbb{R}^{p\times p}$ based on the empirical estimates $λ_i(\hat C_1\hat C_2)$ ($\hat C_a=\frac1{n_a}\sum_{i=1}^{n_a} x_i^{(a)}x_i^{(a){\sf T}}$), when the size $p$ and number $n_a$ of the (zero mean) samples $x_i^{(a)}$ are similar. As a corollary, a consistent estimate of the Wasserstein distance (related to the case $f(t)=\sqrt{t}$) between centered Gaussian distributions is derived. The new estimate is shown to largely outperform the classical sample covariance-based `plug-in' estimator. Based on this finding, a practical application to covariance estimation is then devised which demonstrates potentially significant performance gains with respect to state-of-the-art alternatives.

Measuring large optical reflection matrices of turbid media

We address the problem of estimating the covariance matrix from a complex central angular Gaussian distribution when the number of samples <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">T</i> is less than the size of the observation space <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">M</i> . As regularization is needed, we consider the expected likelihood (EL) approach as a means to set the regularization parameters. The EL principle, originally developed under the Gaussian assumption, relies on some invariance properties of the likelihood ratio (LR). In this paper, we show that the LR, evaluated at the true covariance matrix, has a distribution that only depends on <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">T</i> and <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">M</i> . A simple representation as a product of beta distributed random variables is presented. This paves the way to EL-based regularized covariance matrix estimation, whose effectiveness is shown through simulations.