Real Gaussian Research Articles

Approximate Capacity-Distortion Region of Joint State Sensing and Communication in MIMO Real Gaussian Channels

Approximate Capacity-Distortion Region of Joint State Sensing and Communication in MIMO Real Gaussian Channels

Computation of the Fundamental Limits of Data Compression for Certain Nonstationary ARMA Vector Sources

In the present article, the differential entropy rate and the rate distortion function (RDF) are computed for certain nonstationary real Gaussian autoregressive moving average (ARMA) vector sources.

Cumulants asymptotics for the zeros counting measure of real Gaussian processes

Cumulants asymptotics for the zeros counting measure of real Gaussian processes

On the non-commutative multifractional Brownian motion

We define a non-commutative analogue of a real gaussian Volterra-type multifractional Brownian motion (NC-mfBm for short) and show that its trajectories behave locally like non-commutative fractional Brownian motion. We determine the pointwise Hölder exponent as well as a random matrix approximation of this process.

On the Condition Number of the Shifted Real Ginibre Ensemble

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

Optimization landscape in the simplest constrained random least-square problem

We analyze statistical features of the ‘optimization landscape’ in a random version of one of the simplest constrained optimization problems of the least-square type: finding the best approximation for the solution of a system of M linear equations in N unknowns: ( a k , x ) = b k , k = 1, …, M on the N-sphere x 2 = N. We treat both the N-component vectors a k and parameters b k as independent mean zero real Gaussian random variables. First, we derive the exact expressions for the mean number of stationary points of the least-square loss function in the overcomplete case M > N in the framework of the Kac-Rice approach combined with the random matrix theory for Wishart ensemble. Then we perform its asymptotic analysis as N → ∞ at a fixed α = M/N > 1 in various regimes. In particular, this analysis allows to extract the large deviation function for the density of the smallest Lagrange multiplier λ min associated with the problem, and in this way to find its most probable value. This can be further used to predict the asymptotic mean minimal value of the loss function as N → ∞. Finally, we develop an alternative approach based on the replica trick to conjecture the form of the large deviation function for the density of at N ≫ 1 and any fixed ratio α = M/N > 0. As a by-product, we find the compatibility threshold α c < 1 which is the value of α beyond which a large random linear system on the N-sphere becomes typically incompatible.

PIMA-CT: Physical Model-Aware Cyclic Simulation and Denoising for Ultra-Low-Dose CT Restoration.

A body of studies has proposed to obtain high-quality images from low-dose and noisy Computed Tomography (CT) scans for radiation reduction. However, these studies are designed for population-level data without considering the variation in CT devices and individuals, limiting the current approaches' performance, especially for ultra-low-dose CT imaging. Here, we proposed PIMA-CT, a physical anthropomorphic phantom model integrating an unsupervised learning framework, using a novel deep learning technique called Cyclic Simulation and Denoising (CSD), to address these limitations. We first acquired paired low-dose and standard-dose CT scans of the phantom and then developed two generative neural networks: noise simulator and denoiser. The simulator extracts real low-dose noise and tissue features from two separate image spaces (e.g., low-dose phantom model scans and standard-dose patient scans) into a unified feature space. Meanwhile, the denoiser provides feedback to the simulator on the quality of the generated noise. In this way, the simulator and denoiser cyclically interact to optimize network learning and ease the denoiser to simultaneously remove noise and restore tissue features. We thoroughly evaluate our method for removing both real low-dose noise and Gaussian simulated low-dose noise. The results show that CSD outperforms one of the state-of-the-art denoising algorithms without using any labeled data (actual patients' low-dose CT scans) nor simulated low-dose CT scans. This study may shed light on incorporating physical models in medical imaging, especially for ultra-low level dose CT scans restoration.

A conjectural asymptotic formula for multiplicative chaos in number theory

We investigate a special sequence of random variables $A(N)$ defined by an exponential power series with independent standard complex Gaussians $(X(k))_{k \geq 1}$. Introduced by Hughes, Keating, and O'Connell in the study of random matrix theory, this sequence relates to Gaussian multiplicative chaos (in particular "holomorphic multiplicative chaos'' per Najnudel, Paquette, and Simm) and random multiplicative functions. Soundararajan and Zaman recently determined the order of $\mathbb{E}[|A(N)|]$. By constructing an algorithm to calculate $A(N)$ in $O(N^2 \log N)$ steps, we produce computational evidence that their result can likely be strengthened to an asymptotic result with a numerical estimate for the asymptotic constant. We also obtain similar conclusions when $A(N)$ is defined using standard real Gaussians or uniform $\pm 1$ random variables. However, our evidence suggests that the asymptotic constants do not possess a natural product structure.

Positive Gaussian Kernels also Have Gaussian Minimizers

We study lower bounds on multilinear operators with Gaussian kernels acting on Lebesgue spaces, with exponents below one. We put forward natural conditions when the optimal constant can be computed by inspecting centered Gaussian functions only, and we give necessary and sufficient conditions for this constant to be positive. Our work provides a counterpart to Lieb’s results on maximizers of multilinear operators with real Gaussian kernels, also known as the multidimensional Brascamp-Lieb inequality. It unifies and extends several inverse inequalities.

Simulation of Gaussian random field in a ball

Abstract We address the problem of statistical simulation of a scalar real Gaussian random field inside the unit 3D ball. Two different methods are studied: (i) the method based on the known homogeneous isotropic power spectrum developed by Meschede and Romanowicz [M. Meschede and B. Romanowicz, Non-stationary spherical random media and their effect on long-period mantle waves, Geophys. J. Int. 203 2015, 1605–1625] and (ii) the method based on known radial and angular covariance functions suggested in this work. The first approach allows the extension of the simulation technique to the inhomogeneous or anisotropic case. However, the disadvantage of this approach is the lack of accurate statistical characterization of the results. The accuracy of considered methods is illustrated by numerical tests, including a comparison of the estimated and analytical covariance functions. These methods can be used in many applications in geophysics, geodynamics, or planetary science where the objective is to construct spatial realizations of 3D random fields based on a statistical analysis of observations collected on the sphere or within a spherical region.

Secrecy Capacity of a Gaussian Wiretap Channel With ADCs is Always Positive

We consider a complex Gaussian wiretap channel with finite-resolution analog-to-digital converters (ADCs) at both the legitimate receiver and the eavesdropper. For this channel, we show that a positive secrecy rate is always achievable as long as the channel gains at the legitimate receiver and at the eavesdropper are different, regardless of the quantization levels of the ADCs. For the achievability, we first consider the case of the one-bit ADCs at the legitimate receiver and apply a binary input distribution where the two input points have the same phase when the channel gain at the legitimate receiver is less than that at the eavesdropper, and otherwise the opposite phase. Then the result is generalized for the case of arbitrary finite-resolution ADCs at the legitimate receiver by translating the input distribution appropriately. We also provide numerical lower bounds on the achievable secrecy rates, and analyze the tendency of the secrecy rates according to the channel difference, the power constraint, and the quantization levels for some cases. For the special case of the real Gaussian wiretap channel with one-bit ADCs at both the legitimate receiver and the eavesdropper, we show that our choice of phase does not lose any optimality for the Wyner code.

Non-universal fluctuations of the empirical measure for isotropic stationary fields on S2×R

In this paper, we consider isotropic and stationary real Gaussian random fields defined on S2×R and we investigate the asymptotic behavior, as T→+∞, of the empirical measure (excursion area) in S2×[0,T] at any threshold, covering both cases when the field exhibits short and long memory, that is, integrable and nonintegrable temporal covariance. It turns out that the limiting distribution is not universal, depending both on the memory parameters and the threshold. In particular, in the long memory case a form of Berry’s cancellation phenomenon occurs at zero-level, inducing phase transitions for both variance rates and limiting laws.

On collision of multiple eigenvalues for matrix-valued Gaussian processes

For real symmetric and complex Hermitian Gaussian processes whose values are d×d matrices, we characterize the conditions under which the probability that at least k eigenvalues collide is positive for 2≤k≤d, and we obtain the Hausdorff dimension of the set of collision times.

On the achievable rate of bandlimited continuous-time AWGN channels with 1-bit output quantization

We consider a real continuous-time bandlimited additive white Gaussian noise channel with 1-bit output quantization. On such a channel the information is carried by the temporal distances of the zero-crossings of the transmit signal. We derive an approximate lower bound on the capacity by lower-bounding the mutual information rate for input signals with exponentially distributed zero-crossing distances, sine-shaped transition waveform, and an average power constraint. The focus is on the behavior in the mid-to-high signal-to-noise ratio (SNR) regime above 10 dB. For hard bandlimited channels, the lower bound on the mutual information rate saturates with the SNR growing to infinity. For a given SNR the loss with respect to the unquantized additive white Gaussian noise channel solely depends on the ratio of channel bandwidth and the rate parameter of the exponential distribution. We complement those findings with an approximate upper bound on the mutual information rate for the specific signaling scheme. We show that both bounds are close in the SNR domain of approximately 10–20 dB.

Corank-1 projections and the randomised Horn problem

Let $\hat{\boldsymbol x}$ be a normalised standard complex Gaussian vector, and project an Hermitian matrix $A$ onto the hyperplane orthogonal to $\hat{\boldsymbol x}$. In a recent paper Faraut [Tunisian J. Math. \textbf{1} (2019), 585--606] has observed that the corresponding eigenvalue PDF has an almost identical structure to the eigenvalue PDF for the rank 1 perturbation $A + b \hat{\boldsymbol x} \hat{\boldsymbol x}^\dagger$, and asks for an explanation. We provide this by way of a common derivation involving the secular equations and associated Jacobians. This applies too in related setting, for example when $\hat{\boldsymbol x}$ is a real Gaussian and $A$ Hermitian, and also in a multiplicative setting $A U B U^\dagger$ where $A, B$ are fixed unitary matrices with $B$ a multiplicative rank 1 deviation from unity, and $U$ is a Haar distributed unitary matrix. Specifically, in each case there is a dual eigenvalue problem giving rise to a PDF of almost identical structure.

Fluctuation around the circular law for random matrices with real entries

We extend our recent result [22] on the central limit theorem for the linear eigenvalue statistics of non-Hermitian matrices X with independent, identically distributed complex entries to the real symmetry class. We find that the expectation and variance substantially differ from their complex counterparts, reflecting (i) the special spectral symmetry of real matrices onto the real axis; and (ii) the fact that real i.i.d. matrices have many real eigenvalues. Our result generalizes the previously known special cases where either the test function is analytic [49] or the first four moments of the matrix elements match the real Gaussian [59, 44]. The key element of the proof is the analysis of several weakly dependent Dyson Brownian motions (DBMs). The conceptual novelty of the real case compared with [22] is that the correlation structure of the stochastic differentials in each individual DBM is non-trivial, potentially even jeopardising its well-posedness.

Phase Transitions in Frequency Agile Radar Using Compressed Sensing

FAR has improved anti-jamming performance over traditional pulse-Doppler radars under complex electromagnetic circumstances. To reconstruct the range-Doppler information in FAR, many compressed sensing (CS) methods including standard and block sparse recovery have been applied. In this paper, we study phase transitions of range-Doppler recovery in FAR using CS. In particular, we derive closed-form phase transition curves associated with block sparse recovery and complex Gaussian matrices, based on prior results of standard sparse recovery under real Gaussian matrices. We further approximate the obtained curves with elementary functions of radar and target parameters, facilitating practical applications of these curves. Our results indicate that block sparse recovery outperforms the standard counterpart when targets occupy more than one range cell, which are often referred to as extended targets. Simulations validate the availability of these curves and their approximations in FAR, which benefit the design of the radar parameters.

Condition Numbers for Real Eigenvalues in the Real Elliptic Gaussian Ensemble

We study the distribution of the eigenvalue condition numbers kappa _i=sqrt{ ({mathbf{l}}_i^* {mathbf{l}}_i)({mathbf{r}}_i^* {mathbf{r}}_i)} associated with real eigenvalues lambda _i of partially asymmetric Ntimes N random matrices from the real Elliptic Gaussian ensemble. The large values of kappa _i signal the non-orthogonality of the (bi-orthogonal) set of left {mathbf{l}}_i and right {mathbf{r}}_i eigenvectors and enhanced sensitivity of the associated eigenvalues against perturbations of the matrix entries. We derive the general finite N expression for the joint density function (JDF) {{mathcal {P}}}_N(z,t) of t=kappa _i^2-1 and lambda _i taking value z, and investigate its several scaling regimes in the limit Nrightarrow infty . When the degree of asymmetry is fixed as Nrightarrow infty , the number of real eigenvalues is mathcal {O}(sqrt{N}), and in the bulk of the real spectrum t_i=mathcal {O}(N), while on approaching the spectral edges the non-orthogonality is weaker: t_i=mathcal {O}(sqrt{N}). In both cases the corresponding JDFs, after appropriate rescaling, coincide with those found in the earlier studied case of fully asymmetric (Ginibre) matrices. A different regime of weak asymmetry arises when a finite fraction of N eigenvalues remain real as Nrightarrow infty . In such a regime eigenvectors are weakly non-orthogonal, t=mathcal {O}(1), and we derive the associated JDF, finding that the characteristic tail {{mathcal {P}}}(z,t)sim t^{-2} survives for arbitrary weak asymmetry. As such, it is the most robust feature of the condition number density for real eigenvalues of asymmetric matrices.

Fitting continuum wavefunctions with complex Gaussians: Computation of ionization cross sections.

We implement a full nonlinear optimization method to fit continuum states with complex Gaussians. The application to a set of regular scattering Coulomb functions allows us to validate the numerical feasibility, to explore the range of convergence of the approach, and to demonstrate the relative superiority of complex over real Gaussian expansions. We then consider the photoionization of atomic hydrogen, and ionization by electron impact in the first Born approximation, for which the closed form cross sections serve as a solid benchmark. Using the proposed complex Gaussian representation of the continuum combined with a real Gaussian expansion for the initial bound state, all necessary matrix elements within a partial wave approach become analytical. The successful numerical comparison illustrates that the proposed all-Gaussian approach works efficiently for ionization processes of one-center targets.

Оцiнки розподiлу напiвнорм Гельдера вiд дiйсних стацiонарних гауссових процесiв зi стiйкою кореляцiйною функцiєю

Complex random variables and processes with a vanishing pseudo-correlation are called proper. There is a class of stationary proper complex random processes that have a stable correlation function. In the present article we consider real stationary Gaussian processes with a stable correlation function. It is shown that the trajectories of stationary Gaussian proper complex random processes with zero mean belong to the Orlich space generated by the function $U(x) = e^{x^2/2}-1$. Estimates are obtained for the distribution of semi-norms of sample functions of Gaussian proper complex random processes with a stable correlation function, defined on the compact $\mathbb{T} = [0,T]$, in Hölder spaces.