AR Identification of Latent-Variable Graphical Models

The paper proposes an identification procedure for autoregressive Gaussian stationary stochastic processes under the assumption that the manifest (or observed) variables are nearly independent when conditioned on a limited number of latent (or hidden) variables. The method exploits the sparse plus low-rank decomposition of the inverse of the manifest spectral density and the efficient convex relaxations recently proposed for such decompositions

Factor Analysis of Moving Average Processes

The paper considers an extension of factor analysis to moving average processes. The problem is formulated as a rank minimization of a suitable spectral density. It is shown that it can be adequately approximated via a trace norm convex relaxation

Factor analysis with finite data

Factor analysis aims to describe high dimensional random vectors by means of a small number of unknown common factors. In mathematical terms, it is required to decompose the covariance matrix $\Sigma$ of the random vector as the sum of a diagonal matrix $D$ | accounting for the idiosyncratic noise in the data | and a low rank matrix $R$ | accounting for the variance of the common factors | in such a way that the rank of $R$ is as small as possible so that the number of common factors is minimal. In practice, however, the matrix $\Sigma$ is unknown and must be replaced by its estimate, i.e. the sample covariance, which comes from a finite amount of data. This paper provides a strategy to account for the uncertainty in the estimation of $\Sigma$ in the factor analysis problem.Comment: Draft, the final version will appear in the 56th IEEE Conference on Decision and Control, Melbourne, Australia, 201

An Interpretation of the Dual Problem of the THREE-like Approaches

Spectral estimation can be preformed using the so called THREE-like approach. Such method leads to a convex optimization problem whose solution is characterized through its dual problem. In this paper, we show that the dual problem can be seen as a new parametric spectral estimation problem. This interpretation implies that the THREE-like solution is optimal in terms of closeness to the correlogram over a certain parametric class of spectral densities, enriching in this way its meaningfulness

A Bayesian Approach to Sparse plus Low rank Network Identification

We consider the problem of modeling multivariate time series with parsimonious dynamical models which can be represented as sparse dynamic Bayesian networks with few latent nodes. This structure translates into a sparse plus low rank model. In this paper, we propose a Gaussian regression approach to identify such a model

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

Factor Models with Real Data: a Robust Estimation of the Number of Factors

Factor models are a very efficient way to describe high dimensional vectors of data in terms of a small number of common relevant factors. This problem, which is of fundamental importance in many disciplines, is usually reformulated in mathematical terms as follows. We are given the covariance matrix Sigma of the available data. Sigma must be additively decomposed as the sum of two positive semidefinite matrices D and L: D | that accounts for the idiosyncratic noise affecting the knowledge of each component of the available vector of data | must be diagonal and L must have the smallest possible rank in order to describe the available data in terms of the smallest possible number of independent factors. In practice, however, the matrix Sigma is never known and therefore it must be estimated from the data so that only an approximation of Sigma is actually available. This paper discusses the issues that arise from this uncertainty and provides a strategy to deal with the problem of robustly estimating the number of factors.Comment: arXiv admin note: text overlap with arXiv:1708.0040