Shrinking the Covariance Matrix Using Convex Penalties on the Matrix-Log Transformation

Abstract

For q-dimensional data, penalized versions of the sample covariance matrix are important when the sample size is small or modest relative to q. Since the negative log-likelihood under multivariate normal sampling is convex in ${\Sigma }^{-1}$, the inverse of the covariance matrix, it is common to consider additive penalties which are also convex in ${\Sigma }^{-1}$. More recently, Deng and Tsui and Yu et al. have proposed penalties which are strictly functions of the roots of Σ and are convex in $\text{log}\hspace{0.17em}\Sigma$, but not in ${\Sigma }^{-1}$. The resulting penalized optimization problems, though, are neither convex in $\text{log}\hspace{0.17em}\Sigma$ nor in ${\Sigma }^{-1}$. In this article, however, we show these penalized optimization problems to be geodesically convex in Σ. This allows us to establish the existence and uniqueness of the corresponding penalized covariance matrices. More generally, we show that geodesic convexity in Σ is equivalent to convexity in $\text{log}\hspace{0.17em}\Sigma$ for penalties which are functions of the roots of Σ. In addition, when using such penalties, the resulting penalized optimization problem reduces to a q-dimensional convex optimization problem on the logs of the roots of Σ, which can then be readily solved via Newton’s algorithm. Supplementary materials for this article are available online.

Keywords:

• Geodesic convexity
• Newton–Raphson algorithm
• Penalized likelihood

Supplementary Materials

The supplementary materials include the following: a supplementary manuscript reporting further simulation results, an R-package logconvx for computing the proposed penalized covariance matrices, an Rdata workspace Sim.Rdata for reproducing the simulations reported in the manuscript, and an RData workspace Sonar.Rdata for reproducing the results for the example given in section 7.

Additional information

Funding

Research for both authors was supported in part by the National Science Foundation grants DMS-1407751 and DMS-1812198. Mengxi Yi’s research was also supported in part by the Austrian Science Fund (FWF) under grant P31881-N32 and in part by the Scientific Research Starting Foundation of UIBE.