Abstract

The sparse Cholesky parametrization of the inverse covariance matrix can be interpreted as a Gaussian Bayesian network; however its counterpart, the covariance Cholesky factor, has received, with few notable exceptions, little attention so far, despite having a natural interpretation as a hidden variable model for ordered signal data. To fill this gap, in this paper we focus on arbitrary zero patterns in the Cholesky factor of a covariance matrix. We discuss how these models can also be extended, in analogy with Gaussian Bayesian networks, to data where no apparent order is available. For the ordered scenario, we propose a novel estimation method that is based on matrix loss penalization, as opposed to the existing regression-based approaches. The performance of this sparse model for the Cholesky factor, together with our novel estimator, is assessed in a simulation setting, as well as over spatial and temporal real data where a natural ordering arises among the variables. We give guidelines, based on the empirical results, about which of the methods analysed is more appropriate for each setting.

Highlights

  • The multivariate Gaussian distribution is central in both statistics and machine learning because of its wide applicability and well theoretical behaviour

  • Letting Ω = WWt be its Cholesky decomposition, a zero pattern in the lower triangular matrix W yields the acyclic digraph associated with a Gaussian Bayesian network [6], [7] model, up to a permutation of the variables [8]

  • In this paper we focus on arbitrary zero patterns in the Cholesky factor T of the covariance matrix Σ. We argue how this naturally models scenarios with ordered variables representing noisy inputs from hidden signal sources, see for example [20] where latent brain regions are assumed to influence neural measurements. We discuss how this model could be extended to unordered variables by defining a new Gaussian graphical model over the Cholesky factorization of the covariance matrix

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Summary

INTRODUCTION

The multivariate Gaussian distribution is central in both statistics and machine learning because of its wide applicability and well theoretical behaviour. In this paper we focus on arbitrary zero patterns in the Cholesky factor T of the covariance matrix Σ We argue how this naturally models scenarios with ordered variables representing noisy inputs from hidden signal sources, see for example [20] where latent brain regions are assumed to influence neural measurements. We discuss how this model could be extended to unordered variables by defining a new Gaussian graphical model over the Cholesky factorization of the covariance matrix. Appendices A, B and C are referenced throughout the paper and contain additional material for the interested reader

THEORETICAL PRELIMINARIES
THE CHOLESKY DECOMPOSITION OF A COVARIANCE MATRIX
SPARSE CHOLESKY DECOMPOSITION OF THE
THE REGRESSION INTERPRETATIONS FOR THE COVARIANCE CHOLESKY FACTOR
A HIDDEN VARIABLE MODEL INTERPRETATION
A GRAPHICAL MODEL EXTENSION FOR UNORDERED VARIABLES
EXISTING WORK
PENALIZED GRADIENT-BASED LEARNING OF THE COVARIANCE SPARSE CHOLESKY FACTOR
EXPERIMENTS
DISCUSSION
Findings
CONCLUSIONS AND FUTURE WORK
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