A Log-Det Heuristics for Covariance Matrix Estimation: The Analytic Setup

Enrico Bernardi and Matteo Farnè
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Enrico Bernardi: Dipartimento di Scienze Statistiche, Università di Bologna, 40126 Bologna, Italy
Matteo Farnè: Dipartimento di Scienze Statistiche, Università di Bologna, 40126 Bologna, Italy

Stats, 2022, vol. 5, issue 3, 1-11

Abstract: This paper studies a new nonconvex optimization problem aimed at recovering high-dimensional covariance matrices with a low rank plus sparse structure. The objective is composed of a smooth nonconvex loss and a nonsmooth composite penalty. A number of structural analytic properties of the new heuristics are presented and proven, thus providing the necessary framework for further investigating the statistical applications. In particular, the first and the second derivative of the smooth loss are obtained, its local convexity range is derived, and the Lipschitzianity of its gradient is shown. This opens the path to solve the described problem via a proximal gradient algorithm.

Keywords: nonconvex optimization; local convexity; nuclear norm; covariance matrix; high dimension (search for similar items in EconPapers)
JEL-codes: C1 C10 C11 C14 C15 C16 (search for similar items in EconPapers)
Date: 2022
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