Riemannian Optimization for Variance Estimation in Linear Mixed Models
This addresses a computational bottleneck for statisticians and data scientists working with linear mixed models, though it is an incremental improvement in optimization techniques.
The authors tackled the challenge of variance parameter estimation in linear mixed models by formulating residual maximum likelihood estimation as a Riemannian optimization problem on a manifold, resulting in higher-quality estimates compared to existing methods.
Variance parameter estimation in linear mixed models is a challenge for many classical nonlinear optimization algorithms due to the positive-definiteness constraint of the random effects covariance matrix. We take a completely novel view on parameter estimation in linear mixed models by exploiting the intrinsic geometry of the parameter space. We formulate the problem of residual maximum likelihood estimation as an optimization problem on a Riemannian manifold. Based on the introduced formulation, we give geometric higher-order information on the problem via the Riemannian gradient and the Riemannian Hessian. Based on that, we test our approach with Riemannian optimization algorithms numerically. Our approach yields a higher quality of the variance parameter estimates compared to existing approaches.