Partial correlation analysis is widely used in modern multivariate analysis. However, prior methods lack non-asymptotic guarantees and have slow asymptotic estimation rates. We present a method for simultaneously estimating sparse high-dimensional partial correlation matrices and inverse covariance with non-asymptotic rates. The method is a two-stage estimation which exploits the connection between the inverse covariance matrix and linear regression estimation method while enforcing positive semi-definiteness simultaneously. An efficient proximal splitting algorithm for numerically computing the estimate is also derived, and we provide a fast implementation of the algorithm. The effectiveness of the proposed method is demonstrated on both synthetic and real-world data.
Personalization addresses one of the defining challenges in federated learning: statistical heterogeneity among clients. We show that the dissimilarity between the ideal personalized models of two clients can be upper bounded by the dissimilarity of their distributions. Motivated by this, we propose Karula, a data-driven regularization method that constrains pairwise model dissimilarities between clients according to their distributional differences. Distributional similarity is measured using a surrogate for the 1-Wasserstein distance adapted for the federated setting, allowing the method to adapt to highly complex interrelations between clients that approaches based on hard clustering may fail to recover. We propose an approximate projected stochastic gradient algorithm to solve the resulting constrained optimization problem, and prove that it converges at a rate of $O(1/K)$ to a neighborhood of a stationary point.