Joint Estimation of Precision Matrices in Heterogeneous Populations
This work addresses the challenge of statistical estimation in heterogeneous populations, such as cancer subtypes, with incremental improvements over existing approaches.
The authors tackled the problem of estimating precision matrices from heterogeneous populations by developing a framework that uses Laplacian shrinkage to encourage similarity among subpopulations while allowing differences. Their method achieved consistent estimation and showed advantages in numerical studies and gene expression data from cancer subtypes.
We introduce a general framework for estimation of inverse covariance, or precision, matrices from heterogeneous populations. The proposed framework uses a Laplacian shrinkage penalty to encourage similarity among estimates from disparate, but related, subpopulations, while allowing for differences among matrices. We propose an efficient alternating direction method of multipliers (ADMM) algorithm for parameter estimation, as well as its extension for faster computation in high dimensions by thresholding the empirical covariance matrix to identify the joint block diagonal structure in the estimated precision matrices. We establish both variable selection and norm consistency of the proposed estimator for distributions with exponential or polynomial tails. Further, to extend the applicability of the method to the settings with unknown populations structure, we propose a Laplacian penalty based on hierarchical clustering, and discuss conditions under which this data-driven choice results in consistent estimation of precision matrices in heterogenous populations. Extensive numerical studies and applications to gene expression data from subtypes of cancer with distinct clinical outcomes indicate the potential advantages of the proposed method over existing approaches.