Complex-valued Measures Research Articles

A complex-valued functional brain connectivity descriptor amenable to Riemannian geometry

Objective. We introduce a novel, phase-based, functional connectivity descriptor that encapsulates not only the synchronization strength between distinct brain regions, but also the time-lag between the involved neural oscillations. The new estimator employs complex-valued measurements and results in a brain network sketch that lives on the smooth manifold of Hermitian Positive Definite (HPD) matrices. Approach: Leveraging the HPD property of the proposed descriptor, we adapt a recently introduced dimensionality reduction methodology that is based on Riemannian Geometry and discriminatively detects the recording sites which best reflect the differences in network organization between contrasting recording conditions in order to overcome the problem of high-dimensionality, usually encountered in the connectivity patterns derived from multisite encephalographic recordings. Main results. The proposed framework is validated using an EEG dataset that refers to the challenging problem of differentiating between attentive and passive visual responses. We provide evidence that the reduced connectivity representation facilitates high classification performance and caters for neuroscientific explorations. Significance. Our paper is the very first that introduces an advanced connectivity descriptor that can take advantage of Riemannian geometry tools. The proposed descriptor, that inherently and simultaneously captures both the strength and the corresponding time-lag of the phase synchronization, is the first phase-based descriptor tailored to leverage the benefits of Riemannian geometry.

CONTRACTION SEMIGROUPS ON HYPERGROUP ALGEBRAS

In this paper we develop potential theory on hypergroups and study contrac- tion semigroups on hypergroups, we state and prove a necessary and sufficient condition that the domain of zero-resolvent or the domain of potential operator of {Pt}t>0 is dense in L 2 (K). 1. Introduction and Notation Let K be a commutative locally compact hypergroup. We denote by M (K) the space of all regular complex-valued Borel measures on K ,b yM + (K) the subset of positive measures in M (K), and by δx the Dirac measure at the point x. The closure of any A ⊆ K is denoted by A c .

Davies’ method for heat-kernel estimates: An extension to the semi-elliptic setting

We consider a class of constant-coefficient partial differential operators on a finite-dimensional real vector space which exhibit a natural dilation invariance. Typically, these operators are anisotropic, allowing for different degrees in different directions. The “heat” kernels associated to these so-called positive-homogeneous operators are seen to arise naturally as the limits of convolution powers of complex-valued measures, just as the classical heat kernel appears in the central limit theorem. Building on the functional-analytic approach developed by E. B. Davies for higher-order uniformly elliptic operators with measurable coefficients, we formulate a general theory for (anisotropic) self-adjoint variable-coefficient operators, each comparable to a positive-homogeneous operator, and study their associated heat kernels. Specifically, under three abstract hypotheses, we show that the heat kernels satisfy off-diagonal (Gaussian-type) estimates involving the Legendre-Fenchel transform of the operator’s principle symbol. Our results extend those of E. B. Davies and G. Barbatis and partially extend results of A. F. M. ter Elst and D. Robinson.

Next-generation neural mass and field modeling.

The Wilson-Cowan population model of neural activity has greatly influenced our understanding of the mechanisms for the generation of brain rhythms and the emergence of structured brain activity. As well as the many insights that have been obtained from its mathematical analysis, it is now widely used in the computational neuroscience community for building large-scale in silico brain networks that can incorporate the increasing amount of knowledge from the Human Connectome Project. Here, we consider a neural population model in the spirit of that originally developed by Wilson and Cowan, albeit with the added advantage that it can account for the phenomena of event-related synchronization and desynchronization. This derived mean-field model provides a dynamic description for the evolution of synchrony, as measured by the Kuramoto order parameter, in a large population of quadratic integrate-and-fire model neurons. As in the original Wilson-Cowan framework, the population firing rate is at the heart of our new model; however, in a significant departure from the sigmoidal firing rate function approach, the population firing rate is now obtained as a real-valued function of the complex-valued population synchrony measure. To highlight the usefulness of this next-generation Wilson-Cowan style model, we deploy it in a number of neurobiological contexts, providing understanding of the changes in power spectra observed in electro- and magnetoencephalography neuroimaging studies of motor cortex during movement, insights into patterns of functional connectivity observed during rest and their disruption by transcranial magnetic stimulation, and to describe wave propagation across cortex.

Discrete measures and strict topologies

Let X be a topological Hausdorff space and Bo denote the σ-algebra of Borel sets in X. Then the Banach algebra B(Bo) of all bounded Bo-measurable complex-valued functions on X is a left Banach co(X)-module. The strict topology βd induced on B(Bo) by co(X) coincides with the mixed topology γ[τp,τu], where τu and τp denote the topologies of uniform convergence and pointwise convergence in B(Bo), respectively. A characterization of relative σ(Md(X),B(Bo))-compactness in the Banach space Md(X) of all complex-valued discrete measures is given. We obtain that (B(Bo),βd) is a strongly Mackey space, and hence βd coincides with the Mackey topology τ(B(Bo),Md(X)). It is shown that the space Md(X) has the Schur property, that is, every σ(Md(X),B(Bo))-convergent sequence in Md(X) is norm convergent. Moreover, we study Banach space-valued discrete measures. In particular, we derive a Nikodym type theorem on the setwise sequential convergence of Banach space-valued discrete measures.

Widely linear estimators and adaptive filters for real-valued quantities in complex-valued environments

In this work, mixed real- and complex-valued models are considered. First, the task of estimating a real-valued parameter vector based on complex-valued measurements in a classical set-up is investigated. The application of standard estimators in general results in complex-valued estimates of the real-valued parameter vector. To avoid this systematic error, several widely linear classical estimators that produce real-valued estimates are proposed. The proposed estimators in general outperform standard estimators and they only require half as many complex-valued measurements. Second, adaptive filtering algorithms are proposed for the case that the optimum filter coefficients are real-valued while the input and desired signal are complex-valued. The suggested least mean squares (LMS) and recursive least squares (RLS) type of filters are deployed in a practical application, where they significantly outperform their standard counterparts.

The integral sine addition law

In the present paper we determine, in terms of characters and additive functions, the solutions of the integral functional equation for the sine addition law ∫ G f(xyt)dµ(t) = f(x)g(y) + g(x)f(y), x, y ∈ G, where G is a locally compact Hausdorff group and µ is a regular, compactly supported, complex-valued Borel measure on G. Some consequences of this result and an example are presented.

Complex Correntropy Applied to a Compressive Sensing Problem in an Impulsive Noise Environment

Correntropy is a similarity function capable of extracting high-order statistical information from data. It has been used in different kinds of applications as a cost function to overcome traditional methods in non-Gaussian noise environments. One of the recent applications of correntropy was in the theory of compressive sensing, which takes advantage of sparsity in a transformed domain to reconstruct the signal from a few measurements. Recently, an algorithm called l 0 -MCC was introduced. It applies the Maximum Correntropy Criterion (MCC) in order to deal with a non-Gaussian noise environment in a compressive sensing problem. However, because correntropy was only defined for real-valued data, it was not possible to apply the l 0 -MCC algorithm in a straightforward way to compressive sensing problems dealing with complex-valued measurements. This paper presents a generalization of the l 0 -MCC algorithm to complex-valued measurements. Simulations show that the proposed algorithm can outperform traditional minimization algorithms such as Nesterov's algorithm (NESTA) and the l 0 -Least Mean Square (l 0 -LMS) in the presence of non-Gaussian noise.

Channel edge detection using 2D complex shearlet transform: a case study from the South Caspian Sea

Channels are important sedimentary features in hydrocarbon plays either as targets for drilling or geohazards that should be avoided, depending on burial depth and fluid-fill. Either way, for well design purposes it is important to image channels before drilling. Shearlet transform, as a multi-scale and multi-directional transformation, is capable of detecting anisotropic singularities in two and higher dimensional data. In this study, the complex-valued shearlet-based edge measure was implemented for the aim of channel boundary detection. The method was applied to synthetic seismic time-slices containing channels with different signal-to-noise ratios as well as a real time-slice from the South Caspian Sea. The performance of the shearlet-based algorithm was compared both qualitatively and quantitatively with well known gradient-based edge detectors such as Sobel and Canny, resulting in successfully localising edges and detecting less false positives.

An integral functional equation on groups under two measures

Let G be a locally compact Hausdorff group, let σ be a continuous involutive automorphism on G, and let µ, ν be regular, compactly supported, complex-valued Borel measures on G. We find the continuous solutions f : G → C of the functional equation in terms of continuous characters of G. This equation provides a common generalization of many functional equations (d’Alembert’s, Cauchy’s, Gajda’s, Kannappan’s, Stetkaer’s, Van Vleck’s equations...). So, a large class of functional equations will be solved.

Robust Data Filtering for Estimating Electromechanical Modes of Oscillation via the Multichannel Prony Method

This paper develops a robust estimator of correlation as a data preprocessing stage to the Prony method that is able to suppress white impulsive noise. The method consists of the following steps. First, the bus voltage magnitudes and phase angles are combined to build a set of complex-valued measurements. Second, the outliers of the complex-valued data samples, which are induced by impulsive noise, are identified and suppressed using the iteratively reweighted phase–phase correlator; the latter is a robust estimator of correlation for complex-valued Gaussian processes, which has been extended here to handle outliers in both magnitude and phase angle of voltage phasor measurements. Finally, the classical Prony method is applied on the robustly estimated voltage phase angles. The good performance of the proposed method is demonstrated through simulations carried out on the two-area four-machine system, on the simplified WECC 179-bus system, as well as on real PMU data. Simulation results show that the method is very fast to compute and is compatible with real-time application requirements.

Feynman averaging of semigroups generated by Schrödinger operators

The extension of averaging procedure for operator-valued function is defined by means of the integration of measurable map with respect to complex-valued measure or pseudomeasure. The averaging procedure of one-parametric semigroups of linear operators based on Chernoff equivalence for operator-valued functions is constructed. The initial value problem solutions are investigated for fractional diffusion equation and for Schrödinger equation with relativistic Hamiltonian of free motion. It is established that in these examples the solution of evolutionary equation can be obtained by applying the constructed averaging procedure to the random translation operators in classical coordinate space.

The generalized Van Vleck's equation on locally compact groups

We determine the continuous solutions ʄ, g :G → C of each of the two functional equations ∫G{ʄ,(xyt) – ʄ(σ(y)xt)}dμ(t) = ʄ(x)g(y), x, y ∈ G, ∫G{ʄ,(xyt) – ʄ(σ(y)xt)}dμ(t) = g(x)ʄ(y), x, y ∈ G, where G is a locally compact group, σ is a continuous involutive automorphism on G, and μ is a compactly supported, complex-valued Borel measure on G.

Determining the Dimension of the Improper Signal Subspace in Complex-Valued Data

A complex-valued signal is improper if it is correlated with its complex conjugate. The dimension of the improper signal subspace, i.e., the number of improper components in a complex-valued measurement, is an important parameter and is unknown in most applications. In this letter, we introduce two approaches to estimate this dimension, one based on an information- theoretic criterion and one based on hypothesis testing. We also present reduced-rank versions of these approaches that work for scenarios where the number of observations is comparable to or even smaller than the dimension of the data. Unlike other techniques for determining model orders, our techniques also work in the presence of additive colored noise.

Quantized Spectral Compressed Sensing: Cramer–Rao Bounds and Recovery Algorithms

Efficient estimation of wideband spectrum is of great importance for applications such as cognitive radio. Recently, sub-Nyquist sampling schemes based on compressed sensing have been proposed to greatly reduce the sampling rate. However, the important issue of quantization has not been fully addressed, particularly for high-resolution spectrum and parameter estimation. In this paper, we aim to recover spectrally-sparse signals and the corresponding parameters, such as frequency and amplitudes, from heavy quantizations of their noisy complex-valued random linear measurements, e.g. only the quadrant information. We first characterize the Cramer-Rao bound under Gaussian noise, which highlights the trade-off between sample complexity and bit depth under different signal-to-noise ratios for a fixed budget of bits. Next, we propose a new algorithm based on atomic norm soft thresholding for signal recovery, which is equivalent to proximal mapping of properly designed surrogate signals with respect to the atomic norm that motivates spectral sparsity. The proposed algorithm can be applied to both the single measurement vector case, as well as the multiple measurement vector case. It is shown that under the Gaussian measurement model, the spectral signals can be reconstructed accurately with high probability, as soon as the number of quantized measurements exceeds the order of K log n, where K is the level of spectral sparsity and $n$ is the signal dimension. Finally, numerical simulations are provided to validate the proposed approaches.

Solution of several functional equations on abelian groups with involution

Let G be a locally compact abelian Hausdorff group, let $$\sigma $$ be a continuous involution on G, and let $$\mu ,\nu $$ be regular, compactly supported, complex-valued Borel measures on G. We determine the continuous solutions $$f,g:G\rightarrow {\mathbb {C}}$$ of each of the two functional equations $$\begin{aligned}&\int _{G}f(x+y+t)d\mu (t)+\int _{G}f(x+\sigma (y)+t)d\nu (t)=f(x)g(y),\quad x,y\in G,\\&\int _{G}f(x+y+t)d\mu (t)+\int _{G}f(x+\sigma (y)+t)d\nu (t)=g(x)f(y),\quad x,y\in G, \end{aligned}$$ in terms of characters and additive functions. These equations provides a common generalization of many functional equations such as d’Alembert’s, Cauchy’s, Gajda’s, Kannappan’s, Van Vleck’s, or Wilson’s equations. So, several functional equations will be solved.

Robust DOA Estimation via Sparse Signal Reconstruction With Impulsive Noise

In this letter, direction of arrival (DOA) estimation is considered for an impulsive noise case. The original complex-valued measurement for DOA problem is expressed as an augmented real-valued measurement using the real and imaginary parts. To deal with the impulsive noises, a function approximation technique is used. For the real-valued model, a robust DOA estimation algorithm is proposed based on sparse signal reconstruction. Simulation is given to demonstrate the validity and superiority of our method.

Polynomial Splitting Measures and Cohomology of the Pure Braid Group

We study for each $n$ a one-parameter family of complex-valued measures on the symmetric group $S_n$, which interpolate the probability of a monic, degree $n$, square-free polynomial in $\mathbb{F}_q[x]$ having a given factorization type. For a fixed factorization type, indexed by a partition $\lambda$ of $n$, the measure is known to be a Laurent polynomial. We express the coefficients of this polynomial in terms of characters associated to $S_n$-subrepresentations of the cohomology of the pure braid group $H^{\bullet}(P_n, \mathbb{Q})$. We deduce that the splitting measures for all parameter values $z= -\frac{1}{m}$ (resp. $z= \frac{1}{m}$), after rescaling, are characters of $S_n$-representations (resp. virtual $S_n$-representations.)

Adapting to unknown noise level in sparse deconvolution

In this paper, we study sparse spike deconvolution over the space of complex-valued measures when the input measure is a finite sum of Dirac masses. We introduce a modified version of the Beurling Lasso (BLasso), a semi-definite program that we refer to as the Concomitant Beurling Lasso (CBLasso). This new procedure estimates the target measure and the unknown noise level simultaneously. Contrary to previous estimators in the literature, theory holds for a tuning parameter that depends only on the sample size, so that it can be used for unknown noise level problems. Consistent noise level estimation is standardly proved. As for Radon measure estimation, theoretical guarantees match the previous state-of-the-art results in Super-Resolution regarding minimax prediction and localization. The proofs are based on a bound on the noise level given by a new tail estimate of the supremum of a stationary non-Gaussian process through the Rice method.

Variance analysis of unbiased complex-valued ℓp-norm minimizer

Parameter estimation from noisy complex-valued measurements is a significant topic in various areas of science and engineering. In this aspect, an important goal is finding an unbiased estimator with minimum variance. Therefore, variance analysis of an estimator is desirable and of practical interest. In this paper, we concentrate on analyzing the complex-valued ℓp-norm minimizer with p≥1. Variance formulas for the resultant nonlinear estimators in the presence of three representative bivariate noise distributions, namely, α-stable, Student's t and mixture of generalized Gaussian models, are derived. To guarantee attaining the minimum variance for each noise process, optimum selection of p is studied, in the case of known noise statistics, such as probability density function and corresponding density parameters. All our results are confirmed by simulations and are compared with the Cramér-Rao lower bound.