Non-trivial Covariance Research Articles

Discrete-Time Observations of Brownian Motion on Lie Groups and Homogeneous Spaces: Sampling and Metric Estimation

We present schemes for simulating Brownian bridges on complete and connected Lie groups and homogeneous spaces. We use this to construct an estimation scheme for recovering an unknown left- or right-invariant Riemannian metric on the Lie group from samples. We subsequently show how pushing forward the distributions generated by Brownian motions on the group results in distributions on homogeneous spaces that exhibit a non-trivial covariance structure. The pushforward measure gives rise to new non-parametric families of distributions on commonly occurring spaces such as spheres and symmetric positive tensors. We extend the estimation scheme to fit these distributions to homogeneous space-valued data. We demonstrate both the simulation schemes and estimation procedures on Lie groups and homogenous spaces, including SPD(3)=GL+(3)/SO(3) and S2=SO(3)/SO(2).

Sparse recovery in bounded Riesz systems with applications to numerical methods for PDEs

We study sparse recovery with structured random measurement matrices having independent, identically distributed, and uniformly bounded rows and with a nontrivial covariance structure. This class of matrices arises from random sampling of bounded Riesz systems and generalizes random partial Fourier matrices. Our main result improves the currently available results for the null space and restricted isometry properties of such random matrices. The main novelty of our analysis is a new upper bound for the expectation of the supremum of a Bernoulli process associated with a restricted isometry constant. We apply our result to prove new performance guarantees for the CORSING method, a recently introduced numerical approximation technique for partial differential equations (PDEs) based on compressive sensing.

Optimal designs for a linear-model compositional response

Compositional data play an important role in many disciplines, when the interest is in studying not the total amount but the relative importance or frequency of the involved variables. Due to these proportion/sum constraints, the data belong to a restricted space, the simplex. A special algebraic structure is needed to deal with these kind of data. The use of compositional models has followed an increasing trend during the last years. However, to date not very much has been done about the problem of finding optimal designs for models involving this kind of variables. In this first approach, the application of optimal design theory to models with compositional response is studied, dealing with a possible non-trivial covariance structure between observations. Some analytical results have been obtained, and a clarifying example of application is provided.

Evolution of the fσ8 tension with the Planck15/ΛCDM determination and implications for modified gravity theories

We construct an updated extended compilation of distinct (but possibly correlated) $f\sigma_8(z)$ Redshift Space Distortion (RSD) data published between 2006 and 2018. It consists of 63 datapoints and is significantly larger than previously used similar datasets. After fiducial model correction we obtain the best fit $\Omega_{0m}-\sigma_8$ $\Lambda$CDM parameters and show that they are at a $5\sigma$ tension with the corresponding Planck15/$\Lambda$CDM values. Introducing a nontrivial covariance matrix correlating randomly $20\%$ of the RSD datapoints has no significant effect on the above tension level. We show that the tension disappears (becomes less than $1\sigma$) when a subsample of the 20 most recently published data is used. A partial cause for this reduced tension is the fact that more recent data tend to probe higher redshifts (with higher errorbars) where there is degeneracy among different models due to matter domination. Allowing for a nontrivial evolution of the effective Newton's constant as $G_{\textrm{eff}}(z)/G_{\textrm{N}} = 1 + g_a \left(\frac{z}{1+z}\right)^2 - g_a \left(\frac{z}{1+z}\right)^4$ ($g_a$ is a parameter) and fixing a \plcdm background we find $g_a=-0.91\pm 0.17$ from the full $f\sigma_8$ dataset while the 20 earliest and 20 latest datapoints imply $g_a=-1.28^{+0.28}_{-0.26}$ and $g_a=-0.43^{+0.46}_{-0.41}$ respectively. Thus, the more recent $f\sigma_8$ data appear to favor GR in contrast to earlier data. Finally, we show that the parametrization $f\sigma_8(z)=\lambda \sigma_8 \Omega(z)^\gamma /(1+z)^\beta$ provides an excellent fit to the solution of the growth equation for both GR ($g_a=0$) and modified gravity ($g_a\neq 0$).

Anisotropically Weighted and Nonholonomically Constrained Evolutions on Manifolds

We present evolution equations for a family of paths that results from anisotropically weighting curve energies in non-linear statistics of manifold valued data. This situation arises when performing inference on data that have non-trivial covariance and are anisotropic distributed. The family can be interpreted as most probable paths for a driving semi-martingale that through stochastic development is mapped to the manifold. We discuss how the paths are projections of geodesics for a sub-Riemannian metric on the frame bundle of the manifold, and how the curvature of the underlying connection appears in the sub-Riemannian Hamilton–Jacobi equations. Evolution equations for both metric and cometric formulations of the sub-Riemannian metric are derived. We furthermore show how rank-deficient metrics can be mixed with an underlying Riemannian metric, and we relate the paths to geodesics and polynomials in Riemannian geometry. Examples from the family of paths are visualized on embedded surfaces, and we explore computational representations on finite dimensional landmark manifolds with geometry induced from right-invariant metrics on diffeomorphism groups.

Reconstruction From Anisotropic Random Measurements

Random matrices are widely used in sparse recovery problems, and the relevant properties of matrices with i.i.d. entries are well understood. This paper discusses the recently introduced restricted eigenvalue (RE) condition, which is among the most general assumptions on the matrix, guaranteeing recovery. We prove a reduction principle showing that the RE condition can be guaranteed by checking the restricted isometry on a certain family of low-dimensional subspaces. This principle allows us to establish the RE condition for several broad classes of random matrices with dependent entries, including random matrices with sub-Gaussian rows and nontrivial covariance structure, as well as matrices with independent rows, and uniformly bounded entries.

RIPless compressed sensing from anisotropic measurements

Compressed sensing is the art of reconstructing a sparse vector from its inner products with respect to a small set of randomly chosen measurement vectors. It is usually assumed that the ensemble of measurement vectors is in isotropic position in the sense that the associated covariance matrix is proportional to the identity matrix. In this paper, we establish bounds on the number of required measurements in the anisotropic case, where the ensemble of measurement vectors possesses a non-trivial covariance matrix. Essentially, we find that the required sampling rate grows proportionally to the condition number of the covariance matrix. In contrast to other recent contributions to this problem, our arguments do not rely on any restricted isometry properties (RIP’s), but rather on ideas from convex geometry which have been systematically studied in the theory of low-rank matrix recovery. This allows for a simple argument and slightly improved bounds, but may lead to a worse dependency on noise (which we do not consider in the present paper).

On Ergodic Secrecy Rate for Gaussian MISO Wiretap Channels

A Gaussian multiple-input single-output (MISO) wiretap channel model is considered, where there exists a transmitter equipped with multiple antennas, a legitimate receiver and an eavesdropper, each equipped with a single antenna. We study the problem of finding the optimal input covariance that maximizes the ergodic secrecy rate subject to a power constraint, where only statistical information about the eavesdropper channel is available at the transmitter. This is a non-convex optimization problem that is in general difficult to solve. Existing results address the case in which the eavesdropper or/and legitimate channels have independent and identically distributed Gaussian entries with zero mean and unit variance, i.e., the channels have trivial covariances. This paper addresses the general case in which the eavesdropper and legitimate channels have nontrivial covariances. A set of equations describing the optimal input covariance matrix are proposed along with an algorithm to obtain the solution. Based on this framework, it is shown that when full information on the legitimate channel is available to the transmitter, the optimal input covariance has always rank one. It is also shown that when only statistical information on the legitimate channel is available to the transmitter, the legitimate channel has some general non-trivial covariance and the eavesdropper channel has trivial covariance, the optimal input covariance has the same eigenvectors as the legitimate channel covariance.

Extreme eigenvalue distributions of some complex correlated non-central Wishart and gamma-Wishart random matrices

Let W be a correlated complex non-central Wishart matrix defined through W = X H X , where X is an n × m ( n ≥ m ) complex Gaussian with non-zero mean Υ and non-trivial covariance Σ . We derive exact expressions for the cumulative distribution functions (c.d.f.s) of the extreme eigenvalues (i.e., maximum and minimum) of W for some particular cases. These results are quite simple, involving rapidly converging infinite series, and apply for the practically important case where Υ has rank one. We also derive analogous results for a certain class of gamma-Wishart random matrices, for which Υ H Υ follows a matrix-variate gamma distribution. The eigenvalue distributions in this paper have various applications to wireless communication systems, and arise in other fields such as econometrics, statistical physics, and multivariate statistics.

Relationships between correlation lengths and integral scales for covariance models with more than two parameters

In geostatistical applications, the terms correlation length and range are often used interchangeably and refer to a characteristic covariance length ξ that normalizes the lag distance in the variogram or the covariance model. We present equations that strictly define the correlation length (r c ) and integral range (l c ). We derive analytical expressions for r c and l c of the Whittle–Matern, fluctuation gradient curvature and rational quadratic covariances. For these covariances, we show that the correlation length and integral range for a given model are not fully determined by ξ. We define non-trivial covariance functions, and we formulate an ergodicity index based on l c . We propose using the ergodicity index to compare coarse-grained measures corresponding to non-trivial covariance functions with different parameters. Finally, we discuss potential applications of the proposed covariance models in stochastic subsurface hydrology.

Optimizing multiple-input single-output (miso) communication systems with general gaussian channels: nontrivial covariance and nonzero mean

We consider a narrow-band point-to-point communication system with many (input) transmitters and a single (output) receiver (i.e., a multiple-input single output (MISO) system). We assume the receiver has perfect knowledge of the channel but the transmitter only knows the channel distribution. We focus on two canonical classes of Gaussian channel models: (a) the channel has zero mean with a fixed covariance matrix and (b) the channel has nonzero mean with covariance matrix proportional to the identity. In both cases, we are able to derive simple analytic expressions for the ergodic average and the cumulative distribution function (c.d.f.) of the mutual information for arbitrary input (transmission) signal covariance. With minimal numerical effort, we then determine the ergodic and outage capacities and the corresponding capacity-achieving input signal covariances. Interestingly, we find that the optimal signal covariances for the ergodic and outage cases have very different behavior. In particular, under certain conditions, the outage capacity optimal covariance is a discontinuous function of the parameters describing the channel (such as strength of the correlations or the nonzero mean of the channel).

State space models in process tomography — approximation of state noise covariance

In this paper we consider process tomography in the case of time-varying objects. Especially, we concentrate on the case in which the indirect observations from the system are obtained via electrical impedance tomographic (EIT) measurements and in which the time-evolution of the target can be described by a stochastic convection-diffusion model. We use the state estimation approach to obtain the tomographic reconstructions. The state estimation problem is solved with the fixed-lag Kalman smoother algorithm that is a feasible approach for continuous observation with an insignificant delay in the reconstructions. In particular we focus on the covariance structures associated with state space model. The covariance structures determine the temporal and spatial regularization properties of the algorithm. It is shown that the adoption of nontrivial covariance structures in the evolution model yields good estimates for the time-varying object in such a situation in which stationary reconstructions are completely useless.

On the optimal location of sensors for parametric identification of linear structural systems

An outline of the field of optimal location of sensors for parametric identification of linear structural systems is presented. There are few papers devoted to the case of optimal location of sensors in which the measurements are modeled by a random field with non-trivial covariance function. It is assumed most often that the results of the measurements are statistically independent random variables. In an example the importance of considering the measurements as statistically dependent random variables is shown. The covariance of the model parameters expected to be obtained is investigated with variations in the number and location of sensors. Further, the influence of noise on the optimal location of the sensors is investigated. It is found that the optimal locations of sensors seem to become less sensitive to e.g. the noise-to-signal ratio within increasing number of sensors.