Covariance Estimation in Elliptical Models with Convex Structure

We address structured covariance estimation in Elliptical distribution. We assume it is a priori known that the covariance belongs to a given convex set, e.g., the set of Toeplitz or banded matrices. We consider the General Method of Moments (GMM) optimization subject to these convex constraints. Unfortunately, GMM is still non-convex due to objective. Instead, we propose COCA - a convex relaxation which can be efficiently solved. We prove that the relaxation is tight in the unconstrained case for a finite number of samples, and in the constrained case asymptotically. We then illustrate the advantages of COCA in synthetic simulations with structured Compound Gaussian distributions. In these examples, COCA outperforms competing methods as Tyler's estimate and its projection onto a convex set

Interpolation and Extrapolation of Toeplitz Matrices via Optimal Mass Transport

In this work, we propose a novel method for quantifying distances between Toeplitz structured covariance matrices. By exploiting the spectral representation of Toeplitz matrices, the proposed distance measure is defined based on an optimal mass transport problem in the spectral domain. This may then be interpreted in the covariance domain, suggesting a natural way of interpolating and extrapolating Toeplitz matrices, such that the positive semi-definiteness and the Toeplitz structure of these matrices are preserved. The proposed distance measure is also shown to be contractive with respect to both additive and multiplicative noise, and thereby allows for a quantification of the decreased distance between signals when these are corrupted by noise. Finally, we illustrate how this approach can be used for several applications in signal processing. In particular, we consider interpolation and extrapolation of Toeplitz matrices, as well as clustering problems and tracking of slowly varying stochastic processes

Period Analysis using the Least Absolute Shrinkage and Selection Operator (Lasso)

We introduced least absolute shrinkage and selection operator (lasso) in obtaining periodic signals in unevenly spaced time-series data. A very simple formulation with a combination of a large set of sine and cosine functions has been shown to yield a very robust estimate, and the peaks in the resultant power spectra were very sharp. We studied the response of lasso to low signal-to-noise data, asymmetric signals and very closely separated multiple signals. When the length of the observation is sufficiently long, all of them were not serious obstacles to lasso. We analyzed the 100-year visual observations of delta Cep, and obtained a very accurate period of 5.366326(16) d. The error in period estimation was several times smaller than in Phase Dispersion Minimization. We also modeled the historical data of R Sct, and obtained a reasonable fit to the data. The model, however, lost its predictive ability after the end of the interval used for modeling, which is probably a result of chaotic nature of the pulsations of this star. We also provide a sample R code for making this analysis.Comment: 9 pages, 13 figures, accepted for publication in PAS