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

For <i>q</i>-dimensional data, penalized versions of the sample covariance matrix are important when the sample size is small or modest relative to <i>q</i>. Since the negative log-likelihood under multivariate normal sampling is convex in Σ−1, the inverse of the covariance matrix, it is common to consider additive penalties which are also convex in Σ−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 log Σ, but not in Σ−1. The resulting penalized optimization problems, though, are neither convex in log Σ nor in Σ−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 log Σ for penalties which are functions of the roots of Σ. In addition, when using such penalties, the resulting penalized optimization problem reduces to a <i>q</i>-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.