Gaussian Elements Research Articles

VLSI Design for SVM-Based Speaker Verification System

This brief presents the chip implementation of a support vector machine (SVM)-based speaker verification system. The proposed chip comprises a speaker feature extraction (SFE) module, an SVM module, and a decision module. The SFE module performs autocorrelation analysis, linear predictive coefficient (LPC) extraction, and LPC-to-cepstrum conversion. The SVM module includes a Gaussian kernel unit and a scaling unit. The purpose of the Gaussian kernel unit is first to evaluate the kernel value of a test vector and a support vector. Four Gaussian kernel processing elements (GK-PEs) are designed to process four support vectors simultaneously. Each GK-PE is designed in the pipeline fashion and is capable of performing 2-norm and exponential operations. An enhanced CORDIC architecture is proposed to calculate the exponential value. As well as the Gaussian kernel unit, a scaling unit is also developed for use in the SVM module. The scaling unit is used to perform scaling multiplications and the remaining operations of SVM decision value evaluation. Finally, the decision module accumulates the frame scores that are generated by all of the test frames, and then compare it with a threshold to see if the test utterance is spoken by the claimed speaker. This designed chip is characterized by its high speed and its ability to handle a large number of support vectors in the SVM. The prototype chip is a semicustom chip that is fabricated using Taiwan Semiconductor Manufacturing Company 0.90-nm CMOS technology on a die with a size of ~7.9 × 7.9 mm <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> .

Multi-step prediction of nonlinear Gaussian Process dynamics models with adaptive Gaussian mixtures

This paper presents an adaptive Gaussian Mixture Model (aGMM) formulation for performing multiple-step probabilistic state predictions using a nonparametric Gaussian Process (GP) regression model. The presented prediction algorithm is applicable to any dynamic system that is challenging to model parametrically, but where data is available. Gaussian mixture elements are propagated through the GP by analytically evaluating expectation integrals for the moments of the output distribution. Two metrics are presented and compared for adaptively splitting the initial state distribution into a sum of Gaussians to reduce the effect of nonlinearities on prediction accuracy: (1) an analytical evaluation of the excess kurtosis which measures the non-Gaussianity of the output distribution, and (2) a weighted least-squares regression model which evaluates the local nonlinearity of the GP mapping with respect to the input distribution. In addition, an on-the-fly data selection method is presented to reduce the computational complexity associated with analytically evaluating the higher-order moments of the GP output distribution. The proposed adaptive GP-aGMM formulation is applied to the case of anticipating driver behavior at road intersections using a GP driver behavior model in combination with a parametric vehicle model. Prediction performance for this scenario is evaluated using driving data collected from three human subjects navigating a standard four-way intersection. Results demonstrate that the presented prediction algorithm is capable of accurately capturing multimodal behavior in the GP training data.

Limit Theorems for Two Classes of Random Matrices with Gaussian Entries

In this note, we consider ensembles of random symmetric matrices with Gaussian elements. Assume that $$ \mathbb{E} $$ X ij = 0 and $$ \mathbb{E}{X}_{ij}^2={\sigma}_{ij}^2 $$ We do not assume that all the σ ij are equal. Assuming that the average of the normalized sums of variances in each row converges to one and the Lindeberg condition holds, we prove that the empirical spectral distribution of eigenvalues converges to Wigner’s semicircle law. We also provide an analogue of this result for sample covariance matrices and prove the convergence to the Marchenko–Pastur law. Bibliography: 5 titles.

Diagonally Canonical and Related Gaussian Random Elements

We call a Gaussian random element $\eta$ in a Banach space $X$ with a Schauder basis ${\bf e}=(e_n)$ diagonally canonical (for short, $D$-canonical) with respect to ${\bf e}$ if the distribution of $\eta$ coincides with the distribution of a random element having the form $B\xi$, where $\xi$ is a Gaussian random element in $X$, whose ${\bf e}$-components are stochastically independent and $B: X\to X$ is a continuous linear mapping. In this paper we show that if $X=l_p$, $1\le p<\infty$ and $p\neq2,$ or $X=c_0,$ then there exists a Gaussian random element $\eta$ in $X,$ which is not $D$-canonical with respect to the natural basis of $X$. We derive this result in the case when $X=l_p$, $2< p<\infty$, or $X=c_0$ from the following statement, an analogue which was known earlier only for Banach spaces without an unconditional Schauder basis: if $X=l_p,$ $2<p<\infty,$ or $X=c_0,$ then there exists a Gaussian random element $\eta$ in $X$ such that the distribution of $\eta$ does not coincide with the distribution of the sum of almost surely convergent in $X$ series $\sum_{n=1}^\infty{}x_ng_n$, where $(x_n)$ is an unconditionally summable sequence of elements of $X$ and $(g_n)$ is a sequence of stochastically independent standard Gaussian random variables.

Detection of a sparse submatrix of a high-dimensional noisy matrix

We observe a $N\times M$ matrix $Y_{ij}=s_{ij}+\xi_{ij}$ with $\xi_{ij}\sim {\mathcal {N}}(0,1)$ i.i.d. in $i,j$, and $s_{ij}\in \mathbb {R}$. We test the null hypothesis $s_{ij}=0$ for all $i,j$ against the alternative that there exists some submatrix of size $n\times m$ with significant elements in the sense that $s_{ij}\ge a>0$. We propose a test procedure and compute the asymptotical detection boundary $a$ so that the maximal testing risk tends to 0 as $M\to\infty$, $N\to\infty$, $p=n/N\to0$, $q=m/M\to0$. We prove that this boundary is asymptotically sharp minimax under some additional constraints. Relations with other testing problems are discussed. We propose a testing procedure which adapts to unknown $(n,m)$ within some given set and compute the adaptive sharp rates. The implementation of our test procedure on synthetic data shows excellent behavior for sparse, not necessarily squared matrices. We extend our sharp minimax results in different directions: first, to Gaussian matrices with unknown variance, next, to matrices of random variables having a distribution from an exponential family (non-Gaussian) and, finally, to a two-sided alternative for matrices with Gaussian elements.

Ground-state stability and criticality of two-electron atoms with screened Coulomb potentials using the B-splines basis set

We applied the finite-size scaling method using the B-splines basis set to construct the stability diagram for two-electron atoms with a screened Coulomb potential. The results of this method for two-electron atoms are very accurate in comparison with previous calculations based on Gaussian, Hylleraas and finite-element basis sets. The stability diagram for the screened two-electron atoms shows three distinct regions, i.e. a two-electron region, a one-electron region and a zero-electron region, which correspond to stable, ionized and double ionized atoms, respectively. In previous studies, it was difficult to extend the finite-size scaling calculations to large molecules and extended systems because of the computational cost and the lack of a simple way to increase the number of Gaussian basis elements in a systematic way. Motivated by recent studies showing how one can use B-splines to solve Hartree–Fock and Kohn–Sham equations, this combined finite-size scaling using the B-splines basis set might provide an effective systematic way to treat criticality of large molecules and extended systems. As benchmark calculations, the two-electron systems show the feasibility of this combined approach and provide an accurate reference for comparison.

Observed universality of phase transitions in high-dimensional geometry, with implications for modern data analysis and signal processing

We review connections between phase transitions in high-dimensional combinatorial geometry and phase transitions occurring in modern high-dimensional data analysis and signal processing. In data analysis, such transitions arise as abrupt breakdown of linear model selection, robust data fitting or compressed sensing reconstructions, when the complexity of the model or the number of outliers increases beyond a threshold. In combinatorial geometry, these transitions appear as abrupt changes in the properties of face counts of convex polytopes when the dimensions are varied. The thresholds in these very different problems appear in the same critical locations after appropriate calibration of variables. These thresholds are important in each subject area: for linear modelling, they place hard limits on the degree to which the now ubiquitous high-throughput data analysis can be successful; for robustness, they place hard limits on the degree to which standard robust fitting methods can tolerate outliers before breaking down; for compressed sensing, they define the sharp boundary of the undersampling/sparsity trade-off curve in undersampling theorems. Existing derivations of phase transitions in combinatorial geometry assume that the underlying matrices have independent and identically distributed Gaussian elements. In applications, however, it often seems that Gaussianity is not required. We conducted an extensive computational experiment and formal inferential analysis to test the hypothesis that these phase transitions are universal across a range of underlying matrix ensembles. We ran millions of linear programs using random matrices spanning several matrix ensembles and problem sizes; visually, the empirical phase transitions do not depend on the ensemble, and they agree extremely well with the asymptotic theory assuming Gaussianity. Careful statistical analysis reveals discrepancies that can be explained as transient terms, decaying with problem size. The experimental results are thus consistent with an asymptotic large-n universality across matrix ensembles; finite-sample universality can be rejected.

A detailed derivation of Gaussian orbital-based matrix elements in electron structure calculations

A detailed derivation of analytic solutions is presented for overlap, kinetic, nuclear attraction and electron repulsion integrals involving Cartesian Gaussian-type orbitals. It is demonstrated how s-type orbitals can be used to evaluate integrals with higher angular momentum via the properties of Hermite polynomials and differentiation with respect to non-integration variables.

BER and Outage Probability Approximations for LMMSE Detectors on Correlated MIMO Channels

This paper is devoted to the study of the performance of the linear minimum mean-square error (LMMSE) receiver for (receive) correlated multiple-input multiple-output (MIMO) systems. By the random matrix theory, it is well known that the signal-to-noise ratio (SNR) at the output of this receiver behaves asymptotically like a Gaussian random variable as the number of receive and transmit antennas converge to +infin at the same rate. However, this approximation being inaccurate for the estimation of some performance metrics such as the bit error rate (BER) and the outage probability, especially for small system dimensions, Li proposed convincingly to assume that the SNR follows a generalized gamma distribution which parameters are tuned by computing the first three asymptotic moments of the SNR. In this paper, this technique is generalized to (receive) correlated channels, and closed-form expressions for the first three asymptotic moments of the SNR are provided. To obtain these results, a random matrix theory technique adapted to matrices with Gaussian elements is used. This technique is believed to be simple, efficient, and of broad interest in wireless communications. Simulations are provided, and show that the proposed technique yields in general a good accuracy, even for small system dimensions.

Decomposing pavement surface profiles into a Gaussian sequence

This paper proposes that the non-Gaussian (leptokurtic) nature of pavement surface elevation data is a direct result of the inherent level-type non-stationarity of the process manifested as variations in magnitude or roughness. The hypothesis that random pavement profiles are essentially composed of a sequence of zero-mean random Gaussian processes of varying standard deviations is put forward and tested. This paper introduces a numerical approach for decomposing non-stationary random vibration signals into constituent Gaussian elements by extracting Gaussian component of varying root mean square (RMS) levels from the distribution estimates using a curve fitting algorithm. The validity of the method was tested using a representative set of pavement profiles. The decomposition method presented is significant in that it affords great simplicity for the synthesis of non-stationary pavement profiles which can be achieved without much difficulty when the process is represented by a sequence of Gaussian events.

Gaussian matrix elements in a cylindrical harmonic oscillator basis

We derive a formalism, the separation method, for the efficient and accurate calculation of two-body matrix elements for a Gaussian potential in the cylindrical harmonic-oscillator basis. This formalism is of critical importance for Hartree–Fock and Hartree–Fock–Bogoliubov calculations in deformed nuclei using realistic, finite-range effective interactions between nucleons. The results given here are also relevant for microscopic many-body calculations in atomic and molecular physics, as the formalism can be applied to other types of interactions beyond the Gaussian form. The derivation is presented in great detail to emphasize the methodology, which relies on generating functions. The resulting analytical expressions for the Gaussian matrix elements are checked for speed and accuracy as a function of the number of oscillator shells and against direct numerical integration.

Stability of the dynamics of an asymmetric neural network

We study the stability of the dynamics of a network of $n$ formal neurons interacting through an asymmetric matrix with independent random Gaussian elements of the type introduced by Rajan and Abbott ([1]). The neurons are represented by the values of their electric potentials $x_i, i=1,\cdots,n$. Using the approach developed in a previous paper by us ([6]) we obtain sufficient conditions for diverging synchronized behavior and for stability.

Probability of Local Bifurcation Type from a Fixed Point: A Random Matrix Perspective

Results regarding probable bifurcations from fixed points are presented in the context of general dynamical systems (real, random matrices), time-delay dynamical systems (companion matrices), and a set of mappings known for their properties as universal approximators (neural networks). The eigenvalue spectra is considered both numerically and analytically using previous work of Edelman et. al. Based upon the numerical evidence, various conjectures are presented. The conclusion is that in many circumstances, most bifurcations from fixed points of large dynamical systems will be due to complex eigenvalues. Nevertheless, surprising situations are presented for which the aforementioned conclusion is not general, e.g. real random matrices with Gaussian elements with a large positive mean and finite variance.

Spatial patterns in thunderstorm rainfall events and their coupling with watershed hydrological response

Weather radar systems provide detailed information on spatial rainfall patterns known to play a significant role in runoff generation processes. In the current study, we present an innovative approach to exploit spatial rainfall information of air mass thunderstorms and link it with a watershed hydrological model. Observed radar data are decomposed into sets of rain cells conceptualized as circular Gaussian elements and the associated rain cell parameters, namely, location, maximal intensity and decay factor, are input into a hydrological model. Rain cells were retrieved from radar data for several thunderstorms over southern Arizona. Spatial characteristics of the resulting rain fields were evaluated using data from a dense rain gauge network. For an extreme case study in a semi-arid watershed, rain cells were derived and fed as input into a hydrological model to compute runoff response. A major factor in this event was found to be a single intense rain cell (out of the five cells decomposed from the storm). The path of this cell near watershed tributaries and toward the outlet enhanced generation of high flow. Furthermore, sensitivity analysis to cell characteristics indicated that peak discharge could be a factor of two higher if the cell was initiated just a few kilometers aside.

Better powder diffractometers–choice of beam elements and trade-offs in design

This article re-examines the equations governing resolution on constant wavelength neutron powder diffractometers (PDs) as cast for Gaussian elements. There is a widely accepted choice of collimators and monochromator mosaic to maximise peak intensity for a given resolution on high-resolution PDs. A second distinct “optimisation” was found more recently by an acceptance diagram study. This article shows that these “optimisations” are only two of an infinite number of equivalent element choices. A procedure to find these equivalent configurations is given. This new freedom can be used to overcome technical and other limits on component choice. This article then derives the optimum value of the detector collimation for PDs and discloses a previously unrecognised trade-off in PD design. This collection of new results on PD design may later lead to a true full analytic optimisation of PDs. While this article addresses neutron diffractometers, the results should also apply to modest resolution PDs using other radiation including X-rays. Some of its results have strong implications for all instruments using monochromators.

Isotropic Component Trace Analysis

A new method for analyzing diffusion tensor imaging (DTI) of the brain, based on a recently introduced algorithm, lambda chart analysis (LCA), is presented. Pre-treatment of a given DTI data set with LCA, which effectively segregates isotropic and anisotropic components, allows for total removal of the anisotropic component from the DTI data set. The remaining pure isotropic component can therefore be subjected to further analysis similar to that applied in the trace histogram method. Deconvolution of the trace function yielded 3 Gaussian elements. Remapping of these 3 deconvoluted isotropic elements back onto the 2-dimensional image plane provided anatomical correlates of each element. The algorithm, referred to here as isotropic component trace analysis, can be used as a pictorial analytic tool, as well as a numerical analytical tool, for the noninvasive assessment of isotropic parenchymal components. The presented method provides quantitative indices of certain parenchymal parameters with better clarity than currently available methods. A ready-to-use program, EZ-LCA, for this powerful method is provided (available at http://coe.bri.niigata-u.ac.jp).

Variational Electrostatic Projection (VEP) Methods for Efficient Modeling of the Macromolecular Electrostatic and Solvation Environment in Activated Dynamics Simulations

New methods for the calculation of electrostatic interactions between the active dynamical region and surrounding external solvated macromolecular environment in hybrid quantum mechanical/molecular mechanical (QM/MM) simulations are presented. The variational electrostatic projection (VEP) method, and related variational reverse-mapping procedure (VEP-RVM) utilize an expansion in Gaussian surface elements for representation of electrostatic interactions. The use of a discretized surface that surrounds the active dynamical region greatly reduces the number of interactions with the particles of the external environment. The methods are tested on two catalytic RNA systems: the hammerhead and the hairpin ribozymes. It is shown that with surface elements numbering from 302 to 1202 points the direct VEP and VEP-RVM methods are able to obtain relative force errors in the range of 0.5-0.05% and 0.09-0.0001%, respectively, using a 4.0 A projection buffer. These results are encouraging and provide an essential step in the development of new variational macromolecular solvent-boundary methods for QM/MM calculations of enzyme reactions.

Factoriality of Boejko?Speicher von Neumann Algebras

We study the von Neumann algebra generated by q--deformed Gaussian elements l_i+l_i^* where operators l_i fulfill the q--deformed canonical commutation relations l_i l_j^*-q l_j^* l_i=delta_{ij} for -1<q<1. We show that if the number of generators is finite, greater than some constant depending on q, it is a II_1 factor which does not have the property Gamma. Our technique can be used for proving factoriality of many examples of von Neumann algebras arising from some generalized Brownian motions, both for type II_1 and type III case.

Sample Behavior and Laws of Large Numbers for Gaussian Random Elements

Abstract Criteria for almost sure boundedness and convergence to zero almost surely of Banach space valued independent Gaussian random elements are found. The obtained statements can be viewed as vector-valued versions of the corresponding results due to N. Vakhania. Moreover, from the obtained statements a strong law of large numbers is derived in the form of Yu. V. Prokhorov.

Asymptotic Behavior of Gaussian Samples in Fréchet Spaces

In this article we investigate unnormalized samples of Gaussian random elements in a separable Fréchet space 𝕄. First we describe a connection between shifts of a Gaussian measure μ in a separable Fréchet space and the infinite product of standard normal distributions in ℝ∞, and on the basis of this result we derive the so‐called self‐sufficient expansion for Gaussian random elements in a Fréchet space. Moreover, we find lower bounds for the Gaussian measure μ of shifted balls in 𝕄 and estimate the metric entropy of balls in the Hilbert space ℍ ⊂ 𝕄 which generates μ. Finally, applying the Brunn–Minkowski inequality we prove a kind of the logarithmic law of large numbers. The last result is an extension of the analogous theorem obtained by Goodman (Characteristics of normal samples. Ann. Probab. 1988, 16, 1281–1290), for a sequence of Gaussian random elements in a separable Banach space.