Showing papers on "Complex normal distribution published in 2021"
TL;DR: The first theoretical estimation of the measurement number for stably recovering complex sparse signals from complex Gaussian quadratic measurements is presented, establishing that Gaussian random measurements satisfy the restricted isometry property over rank-$2$ and sparse matrices with high probability.
11 citations
TL;DR: In this article, the authors studied the spectra of N × N Toeplitz band matrices perturbed by small complex Gaussian random matrices and proved a probabilistic Weyl law, which provides a precise asymptotic formula for the number of eigenvalues in certain domains.
11 citations
TL;DR: In this article, the authors consider m independent random rectangular matrices whose entries are independent and identically distributed standard complex Gaussian random variables and assume the product of the m rectangular matrix is an n by n square matrix.
4 citations
TL;DR: In this article, the authors proposed to use the propagation of light in disordered photonic lattices as a random projection that preserves distances between a set of projected vectors, which can be used as a simple and powerful integrated dimension reduction stage to greatly reduce the burden of a subsequent neural computing stage.
Abstract: It is proposed that the propagation of light in disordered photonic lattices can be harnessed as a random projection that preserves distances between a set of projected vectors. This mapping is enabled by the complex evolution matrix of a photonic lattice with diagonal disorder, which turns out to be a random complex Gaussian matrix. Thus, by collecting the output light from a subset of the waveguide channels, one can perform an embedding from a higher-dimension to a lower-dimension space that respects the Johnson-Lindenstrauss lemma and nearly preserves the Euclidean distances. It is discussed that distance-preserving random projection through photonic lattices requires intermediate disorder levels that allow diffusive spreading of light from a single channel excitation, as opposed to strong disorder which initiates the localization regime. The proposed scheme can be utilized as a simple and powerful integrated dimension reduction stage that can greatly reduce the burden of a subsequent neural computing stage.
4 citations
TL;DR: In this paper, the authors consider the problem of estimating a real low-complexity signal from the phase of complex random measurements, and they show that the signal direction can be recovered with high probability and up to global unknown amplitude if the sensing matrix is a complex Gaussian random matrix and the number of measurements is large compared to the complexity of the signal space.
Abstract: We consider the question of estimating a real low-complexity signal (such as a sparse vector or a low-rank matrix) from the phase of complex random measurements. We show that in this phase-only compressive sensing (PO-CS) scenario, we can perfectly recover such a signal with high probability and up to global unknown amplitude if the sensing matrix is a complex Gaussian random matrix and the number of measurements is large compared to the complexity level of the signal space. Our approach proceeds by recasting the (non-linear) PO-CS scheme as a linear compressive sensing model built from a signal normalization constraint, and a phase-consistency constraint imposing any signal estimate to match the observed phases in the measurement domain. Practically, stable and robust estimation of the signal direction is achieved from any instance optimal algorithm of the compressive sensing literature (such as basis pursuit denoising). This is ensured by proving that the matrix associated with this equivalent linear model satisfies with high probability the restricted isometry property under the above condition on the number of measurements. We finally observe experimentally that robust signal direction recovery is reached at about twice the number of measurements needed for signal recovery in compressive sensing.
3 citations
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TL;DR: In this paper, the authors investigated the asymptotic distribution of the maximum of a frequency smoothed estimate of the spectral coherence of a M-variate complex Gaussian time series with mutually independent components when the dimension M and the number of samples N both converge to infinity.
Abstract: We investigate the asymptotic distribution of the maximum of a frequency smoothed estimate of the spectral coherence of a M-variate complex Gaussian time series with mutually independent components when the dimension M and the number of samples N both converge to infinity. If B denotes the smoothing span of the underlying smoothed periodogram estimator, a type I extreme value limiting distribution is obtained under the rate assumptions M N $\rightarrow$ 0 and M B $\rightarrow$ c $\in$ (0, +$\infty$). This result is then exploited to build a statistic with controlled asymptotic level for testing independence between the M components of the observed time series. Numerical simulations support our results.
2 citations
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TL;DR: In this article, it was shown that the randomized Kaczmarz method with a good initialization converges linearly to the target solution in expectation with high probability, up to a global phase.
Abstract: A randomized Kaczmarz method was recently proposed for phase retrieval, which has been shown numerically to exhibit empirical performance over other state-of-the-art phase retrieval algorithms both in terms of the sampling complexity and in terms of computation time. While the rate of convergence has been studied well in the real case where the signals and measurement vectors are all real-valued, there is no guarantee for the convergence in the complex case. In fact, the linear convergence of the randomized Kaczmarz method for phase retrieval in the complex setting is left as a conjecture by Tan and Vershynin. In this paper, we provide the first theoretical guarantees for it. We show that for random measurements $\mathbf{a}_j \in \mathbb{C}^n, j=1,\ldots,m $ which are drawn independently and uniformly from the complex unit sphere, or equivalent are independent complex Gaussian random vectors, when $m \ge Cn$ for some universal positive constant $C$, the randomized Kaczmarz scheme with a good initialization converges linearly to the target solution (up to a global phase) in expectation with high probability. This gives a positive answer to that conjecture.
2 citations
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TL;DR: In this article, the authors proposed a new encoding technique, namely complex encoding, that provides a symmetric representation of categorical values in the complex plane. But, the complexity of complex encoding does not scale well with the dimensionality of the data.
Abstract: A salient problem in machine learning is transforming categorical variables into efficient numerical features. This focus is warranted due to the ubiquity of categorical data in real-world applications but, on the contrary, the development of many machine learning methods based on the assumption of having numerical variables. Perhaps the most popular existing categorical to numerical conversion techniques include one-hot encoding, thermometer encoding, and ordinal (integer) encoding. One-hot and thermometer encodings become computationally inefficient and may lead to ill-conditioning for high-cardinality categorical variables as they create high-dimensional data matrices. As ordinal encoding does not change data dimensionality, it is significantly more memory-efficient; however, the spurious ordering induced by ordinal encoding among unordered categorical values can hamstring the performance of constructed predictive models. In this paper, we propose a new encoding technique, namely complex encoding, that provides a symmetric representation of categorical values in the complex plane. To show the efficacy of the complex encoding in terms of error rate and memory usage, we conducted a set of numerical experiments using real datasets and the family of linear discriminant functions for complex Gaussian distributions. Empirical results show that not only complex encoding avoids the ill-conditioning problem of one-hot and thermometer encodings, it can generally lead to a comparable or higher classification accuracy with respect to others (when they are applicable) at the expense of only about two-fold increase in memory usage with respect to ordinal encoding.
2 citations
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TL;DR: In this article, it was shown that the zero set of an infinite sequence of complex Gaussian random variables such that their covariance matrix is invertible and its inverse is a Toeplitz matrix constitutes a determinantal point process with the same distribution as the case of i.i.d variables studied by Peres and Virag.
Abstract: Given a sequence $(\xi_n)$ of standard i.i.d complex Gaussian random variables, Peres and Virag (in the paper ``Zeros of the i.i.d. Gaussian power series: a conformally invariant determinantal process'' {\it Acta Math.} (2005) 194, 1-35) discovered the striking fact that the zeros of the random power series $f(z) = \sum_{n=1}^\infty \xi_n z^{n-1}$ in the complex unit disc $\mathbb{D}$ constitute a determinantal point process. The study of the zeros of the general random series
$f(z)$ where the restriction of independence is relaxed upon the random variables $(\xi_n)$ is an important open problem. This paper proves that if $(\xi_n)$ is an infinite sequence of complex Gaussian random variables such that their covariance matrix is invertible and its inverse is a Toeplitz matrix, then the zero set of $f(z)$ constitutes a determinantal point process with the same distribution as the case of i.i.d variables studied by Peres and Virag. The arguments are based on some interplays between Hardy spaces and reproducing kernels. Illustrative examples are constructed from classical Toeplitz matrices and the classical fractional Gaussian noise.
2 citations
TL;DR: In this paper, the authors considered a very flexible financial model to construct comonotonic lower convex order bounds in approximating the distribution of the sums of dependent log skew normal random variables.
Abstract: The classical works in finance and insurance for modeling asset returns is the Gaussian model. However, when modeling complex random phenomena, more flexible distributions are needed which are beyond the normal distribution. This is because most of the financial and economic data are skewed and have ＂fat tails＂. Hence symmetric distributions like normal or others may not be good choices while modeling these kinds of data. Flexible distributions like skew normal distribution allow robust modeling of high-dimensional multimodal and asymmetric data. In this paper, we consider a very flexible financial model to construct comonotonic lower convex order bounds in approximating the distribution of the sums of dependent log skew normal random variables. The dependence structure of these random variables is based on a recently developed generalized multivariate skew normal distribution, known as unified skew normal distribution. The approximations are used to calculate the risk measure related to the distribution of terminal wealth. The accurateness of the approximation is investigated numerically. Results obtained from our methods are competitive with a more time consuming method known as Monte Carlo method.
1 citations
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TL;DR: In this paper, the authors considered the case where the dimension of the product matrix goes to infinity and the entries of the matrix are independent and identically distributed standard complex Gaussian random variables.
Abstract: In this paper, we consider $m$ independent random rectangular matrices whose entries are independent and identically distributed standard complex Gaussian random variables and assume the product of the $m$ rectangular matrices is an $n$ by $n$ square matrix. We study the limiting empirical spectral distributions of the product where the dimension of the product matrix goes to infinity, and $m$ may change with the dimension of the product matrix and diverge. We give a complete description for the limiting distribution of the empirical spectral distributions for the product matrix and illustrate some examples.
01 Apr 2021
TL;DR: In this article, the asymptotic behavior of the distribution of the squared singular values of the sample autocovariance matrix between the past and the future of a high-dimensional complex Gaussian uncorrelated se...
Abstract: The asymptotic behavior of the distribution of the squared singular values of the sample autocovariance matrix between the past and the future of a high-dimensional complex Gaussian uncorrelated se...
TL;DR: In this paper, a Gaussian approach is proposed to calculate partial cross-sections and asymmetry parameters for molecular photoionization, where the optimal sets of complex Gaussian-type orbitals (cGTOs) are obtained by nonlinear optimization, to best fit sets of Coulomb or distorted continuum wave functions for relevant orbital quantum numbers.
Abstract: We develop and implement a Gaussian approach to calculate partial cross-sections and asymmetry parameters for molecular photoionization. Optimal sets of complex Gaussian-type orbitals (cGTOs) are first obtained by nonlinear optimization, to best fit sets of Coulomb or distorted continuum wave functions for relevant orbital quantum numbers. This allows us to represent the radial wavefunction for the outgoing electron with accurate cGTO expansions. Within a time-independent partial wave approach, we show that all the necessary transition integrals become analytical, in both length and velocity gauges, thus facilitating the numerical evaluation of photoionization observables. Illustrative results, presented for NH3 and H2 O within a one-active-electron monocentric model, validate numerically the proposed strategy based on a complex Gaussian representation of continuum states.
07 May 2021
TL;DR: This study first derive the Rao detector under the assumption of known covariance matrix of the clutter, and then reconstruct it by AR parameters resorting to the matrix factorization, which proves the effectiveness in small data record case of the newly derived detector.
Abstract: Our study considers the adaptive detection of point targets in compound Gaussian clutter which is in possession of unknown covariance matrix. To overcome the performance degradation problem which is aroused by the limited number of training data, the autoregressive (AR) process is used for the modeling of the speckle component. We first derive the Rao detector under the assumption of known covariance matrix of the clutter, and then reconstruct it by AR parameters resorting to the matrix factorization. Meanwhile, the newly derived detector is proved asymptotically constant false alarm rate with respect to the clutter covariance matrix. Finally, simulation results have confirmed the effectiveness in small data record case of the newly derived detector.
TL;DR: In this paper, the authors proposed to use the propagation of light in disordered photonic lattices as a random projection that preserves distances between a set of projected vectors, which can be utilized as a simple and powerful integrated dimension reduction stage that can greatly reduce the burden of subsequent neural computation stage.
Abstract: It is proposed that the propagation of light in disordered photonic lattices can be harnessed as a random projection that preserves distances between a set of projected vectors. This mapping is enabled by the complex evolution matrix of a photonic lattice with diagonal disorder, which turns out to be a random complex Gaussian matrix. Thus, by collecting the output light from a random subset of the waveguide channels, one can perform an embedding from a higher- to a lower-dimensional space that respects the Johnson–Lindenstrauss lemma and nearly preserves the Euclidean distances. The distance-preserving random projection through photonic lattices requires intermediate disorder levels that allow diffusive propagation of light. The proposed scheme can be utilized as a simple and powerful integrated dimension reduction stage that can greatly reduce the burden of a subsequent neural computation stage.
27 Sep 2021
TL;DR: In this article, the authors derived the Cramer-Rao lower bound for Bayesian estimation of a row-sparse matrix based on quantized low-dimensional multiple measurement vectors (MMV) and derived the joint probability distribution of the observables, estimand and the hyperparameters.
Abstract: We derive the Cramer-Rao lower bound (CRLB) on the mean squared error for Bayesian estimation of a row-sparse matrix based on quantized low-dimensional multiple measurement vectors (MMV). We impose a two-stage hierarchical circularly symmetric complex Gaussian prior parameterized by a diagonal precision hyperparameter matrix on the estimand. The precision hyperparameters themselves assume a non-informative conjugate hyperprior that induces a heavy-tailed Student’s t marginalized prior, which in turn promotes a sparse solution. We derive the joint probability distribution of the observables, estimand and the hyperparameters, and the associated Fisher information matrix (FIM). Due to the analytical intractability in computing the FIM, we resort to Monte Carlo numerical methods that closely approximate the FIM. We demonstrate that the CRLB is tight by considering a variational Bayes orthogonal frequency division multiplexing (OFDM) channel estimator in massive multiple-input multiple-output (MIMO) wireless communication systems with low-resolution analog-to-digital converters as an application and provide further insights on the numerical results.
TL;DR: Four novel non-uniform quantizers are proposed to reduce the mean square error distortion and, compared to the existing high-rate golden quantizer, entropy-coded high- rate golden quantized, and (non)-uniform polar/rectangular quantizers, Bd1-GQ and EC Bd2-GZ achieve lower mean square errors.
Abstract: In this letter, we focus on numerical solutions to the problem of quantizing circularly symmetric complex Gaussian random variables from open symmetrized bidiscs in $\mathbb {C}^{2}$ . To that end, we leverage techniques from the complex variable analysis, especially the symmetrization map. Four novel non-uniform quantizers, i.e. Bd1/2-GQ and EC Bd1/2-GQ, are proposed to reduce the mean square error distortion. Compared to the existing high-rate golden quantizer, entropy-coded high-rate golden quantizer, and (non)-uniform polar/rectangular quantizers, Bd1-GQ and EC Bd1-GQ achieve lower mean square error distortion. Moreover, the EC Bd2-GQ reaches better performance than the polar- and rectangular- quantizers.
TL;DR: In this article, a Gaussian approach is proposed to calculate partial cross-sections and asymmetry parameters for molecular photoionization, where the optimal sets of complex Gaussian-type orbitals (cGTOs) are first obtained by nonlinear optimization, to best fit sets of Coulomb or distorted continuum wave functions for relevant orbital quantum numbers.
Abstract: We develop and implement a Gaussian approach to calculate partial cross-sections and asymmetry parameters for molecular photoionization. Optimal sets of complex Gaussian-type orbitals (cGTOs) are first obtained by non-linear optimization, to best fit sets of Coulomb or distorted continuum wave functions for relevant orbital quantum numbers. This allows us to represent the radial wavefunction for the outgoing electron with accurate cGTO expansions. Within a time-independent partial wave approach, we show that all the necessary transition integrals become analytical, in both length and velocity gauges, thus facilitating the numerical evaluation of photoionization observables. Illustrative results, presented for NH3 and H2O within a one-active-electron monocentric model, validate numerically the proposed strategy based on a complex Gaussian representation of continuum states.
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TL;DR: In this article, the empirical autocovariance matrix at a given non-zero time lag based on observations from a multivariate complex Gaussian stationary time series is studied.
Abstract: Consider the empirical autocovariance matrix at a given non-zero time lag based on observations from a multivariate complex Gaussian stationary time series. The spectral analysis of these autocovariance matrices can be useful in certain statistical problems, such as those related to testing for white noise. We study the behavior of their spectral measures in the asymptotic regime where the time series dimension and the observation window length both grow to infinity, and at the same rate. Following a general framework in the field of the spectral analysis of large random non-Hermitian matrices, at first the probabilistic behavior of the small singular values of the shifted versions of the autocovariance matrix are obtained. This is then used to infer about the large sample behaviour of the empirical spectral measure of the autocovariance matrices at any lag. Matrix orthogonal polynomials on the unit circle play a crucial role in our study.
TL;DR: The concept of threshold array signal-to-noise ratio (ASNR), defined as the minimal SNR at which specific high-resolution algorithms are able to resolve two closely spaced far-field sources, allows to quantify and to compare sensors array performance in localizing remote targets as discussed by the authors.
TL;DR: In the presence of 32-bit floating point single precision processing, the proposed model is more stable in the least squares (LS) parameter estimation and yield better adjacent channel power rejection (ACPR) and normalized mean square error (NMSE) improvement when compared with the conventional model and the Legendre model.
Abstract: Two-dimensional (2-D) polynomial models are widely used in digital predistortion (DPD) design for dual-band power amplifier (PA) linearization. However, the conventional polynomial model exhibits numerical instabilities when high-order nonlinearities are included. To solve this problem, a closed-form 2-D orthogonal polynomial DPD model is deduced under the assumption of complex Gaussian processes. Compared with the Legendre orthogonal polynomial for uniform distribution, the complex Gaussian assumption is more coincident with practical communication signals. Cutting down the condition number of the input correlation matrix is the most important objective to deduce the new model, and the experiment results show that the condition number of the new model can be significantly decreased both for the Gaussian signal and the practical OFDM signal. The predistortion performance of the OFDM signal is also investigated. In the presence of 32-bit floating point single precision processing, the proposed model is more stable in the least squares (LS) parameter estimation and yield better adjacent channel power ratio (ACPR) and normalized mean square error (NMSE) improvement when compared with the conventional model and the Legendre model. At the same time, the proposed model has fewer polynomial coefficients compared with the Legendre model and accordingly reduce the complexity of coefficient estimation.
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TL;DR: In this article, a set of N-body simulation data was used to verify that the one-point distribution functions of the dark matter momentum divergence and density fields closely follow complex Gaussian distributions.
Abstract: The Fourier transformation is an effective and efficient operation of Gaussianization at the one-point level. Using a set of N-body simulation data, we verified that the one-point distribution functions of the dark matter momentum divergence and density fields closely follow complex Gaussian distributions. Statistical theories about the one-point distribution function of the quotient of two complex Gaussian random variables are introduced and applied to model one-point statistics about the growth of individual Fourier mode of the dark matter density field, namely the mode-dependent growth function and the mode growth rate, which can be obtained by the ratio of two Fourier transformed cosmic fields. Our simulation results proved that the models based on the Gaussian approximation are impressively accurate, and our analysis revealed many interesting aspects about the growth of dark matter's density fluctuation in Fourier space.
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TL;DR: In this article, a regularized convex relaxation (RCR) detector was proposed for complex-valued data detection in massive multiple-input multiple-output (MIMO) systems and asymptotic expressions for its mean square error and symbol error probability were derived.
Abstract: In this work, we study complex-valued data detection performance in massive multiple-input multiple-output (MIMO) systems. We focus on the problem of recovering an $n$-dimensional signal whose entries are drawn from an arbitrary constellation $\mathcal{K} \subset \mathbb{C}$ from $m$ noisy linear measurements, with an independent and identically distributed (i.i.d.) complex Gaussian channel. Since the optimal maximum likelihood (ML) detector is computationally prohibitive for large dimensions, many convex relaxation heuristic methods have been proposed to solve the detection problem. In this paper, we consider a regularized version of this convex relaxation that we call the regularized convex relaxation (RCR) detector and sharply derive asymptotic expressions for its mean square error and symbol error probability. Monte-Carlo simulations are provided to validate the derived analytical results.
26 Mar 2021
TL;DR: In this paper, the adaptive detection of range-spread targets in the context of compound Gaussian clutter, which is in possession of unknown covariance matrix, is mainly focused on the problem of performance degradation which is principally triggered by the limitation of training data.
Abstract: In this study, we mainly focus on the adaptive detection of range-spread targets in the context of compound Gaussian clutter, which is in possession of unknown covariance matrix. With the purpose to overcome the problem of performance degradation which is principally triggered by the limitation of training data number, the autoregressive process is applied to model the speckle component. Firstly, the form of Rao test is derived under the assumption of known covariance matrix of the clutter, afterwards the covariance matrix is reconstructed by AR parameters resorting to matrix factorization. The newly derived detector is proved asymptotically constant false alarm rate in respect of the clutter covariance matrix, and the simulation results have demonstrated the effectiveness of the new detector.
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TL;DR: In this article, the mean of the Stieltjes transform of the empirical spectral distribution of any self-adjoint noncommutative polynomial in a Wigner matrix and a deterministic diagonal matrix was investigated, and the convergence to a centred complex Gaussian process whose covariance is expressed in terms of operator-valued subordination functions was obtained.
Abstract: We investigate the fluctuations around the mean of the Stieltjes transform of the empirical spectral distribution of any selfadjoint noncommutative polynomial in a Wigner matrix and a deterministic diagonal matrix. We obtain the convergence in distribution to a centred complex Gaussian process whose covariance is expressed in terms of operator-valued subordination functions.
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TL;DR: In this article, the authors studied permanents of random i.i.d. complex Gaussian matrices, and more generally, submatrices of random unitary matrices.
Abstract: Computing the distribution of permanents of random matrices has been an outstanding open problem for several decades. In quantum computing, "anti-concentration" of this distribution is an unproven input for the proof of hardness of the task of boson-sampling. We study permanents of random i.i.d. complex Gaussian matrices, and more broadly, submatrices of random unitary matrices. Using a hybrid representation-theoretic and combinatorial approach, we prove strong lower bounds for all moments of the permanent distribution. We provide substantial evidence that our bounds are close to being tight and constitute accurate estimates for the moments. Let $U(d)^{k\times k}$ be the distribution of $k\times k$ submatrices of $d\times d$ random unitary matrices, and $G^{k\times k}$ be the distribution of $k\times k$ complex Gaussian matrices. (1) Using the Schur-Weyl duality (or the Howe duality), we prove an expansion formula for the $2t$-th moment of $|Perm(M)|$ when $M$ is drawn from $U(d)^{k\times k}$ or $G^{k\times k}$. (2) We prove a surprising size-moment duality: the $2t$-th moment of the permanent of random $k\times k$ matrices is equal to the $2k$-th moment of the permanent of $t\times t$ matrices. (3) We design an algorithm to exactly compute high moments of the permanent of small matrices. (4) We prove lower bounds for arbitrary moments of permanents of matrices drawn from $G^{ k\times k}$ or $U(k)$, and conjecture that our lower bounds are close to saturation up to a small multiplicative error. (5) Assuming our conjectures, we use the large deviation theory to compute the tail of the distribution of log-permanent of Gaussian matrices for the first time. (6) We argue that it is unlikely that the permanent distribution can be uniquely determined from the integer moments and one may need to supplement the moment calculations with extra assumptions to prove the anti-concentration conjecture.
TL;DR: In this article, the expected density of zeros of derivatives of Gaussian random polynomials on plane domains of C is studied and the main result is that the density of any k-th derivatives of R N tends to the equilibrium measure of the domain.
06 Jun 2021
TL;DR: In this article, the optimal maximum likelihood (ML) detector was developed for cases in which the receiver has full channel state information (CSI), full channel distribution information (CDI), or partial CDI about the transmitter channel.
Abstract: We consider a scenario in which a transmitter sends complex multidimensional symbols to a receiver in the presence of a proactive continuous jammer emitting a zero-mean complex Gaussian signal over an unknown complex Gaussian channel. The complex Gaussian signal transmitted over the unknown complex Gaussian channel induces a non-Gaussian signal at the receiver. We develop the optimal maximum likelihood (ML) detector for cases in which the receiver has full channel state information (CSI), full channel distribution information (CDI), or partial CDI about the transmitter channel. The jammer CDI is either partially or fully available at the receiver. We identify cases in which the non-Gaussian signals resulting from the jammer's transmission can be approximated by Gaussian signals to reduce the computational cost without compromising optimality of detection. Furthermore, we identify cases in which the Gaussian approximation ML detector is not equivalent to the exact ML detector. In these cases, we show that the advantage of the exact ML detector over the Gaussian approximation one can be significant.