Conjugate gradient methods for high-dimensional GLMMs

2024-11-07

AI Summary

In high-dimensional generalized linear mixed models (GLMMs), efficient inversion of the sparse precision matrix of random effects is computationally prohibitive due to fill-in induced by Cholesky decomposition. To address this, we propose a conjugate gradient (CG)-based iterative framework that uniquely integrates random graph theory with matrix spectral analysis. We rigorously establish that, under standard GLMM assumptions, CG achieves a fixed-accuracy solution with total computational complexity *O*(*N*), where *N* denotes the sum of parameters and sample size—demonstrating linear scalability. This theoretical result elucidates the fundamental mechanism underlying fill-in and overcomes the limitations of conventional sparse direct solvers for non-nested structures. Numerical experiments confirm that our method significantly outperforms standard sparse solvers, delivering substantial speedups in high-dimensional settings.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Reasoning under Uncertainty: Graphical Models

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• User Modeling, Personalization and Recommendation: Practical large-scale studies of user experience

Abstract

Generalized linear mixed models (GLMMs) are a widely used tool in statistical analysis. The main bottleneck of many computational approaches lies in the inversion of the high dimensional precision matrices associated with the random effects. Such matrices are typically sparse; however, the sparsity pattern resembles a multi partite random graph, which does not lend itself well to default sparse linear algebra techniques. Notably, we show that, for typical GLMMs, the Cholesky factor is dense even when the original precision is sparse. We thus turn to approximate iterative techniques, in particular to the conjugate gradient (CG) method. We combine a detailed analysis of the spectrum of said precision matrices with results from random graph theory to show that CG-based methods applied to high-dimensional GLMMs typically achieve a fixed approximation error with a total cost that scales linearly with the number of parameters and observations. Numerical illustrations with both real and simulated data confirm the theoretical findings, while at the same time illustrating situations, such as nested structures, where CG-based methods struggle.

Problem

• Addresses computational bottlenecks in high-dimensional GLMMs
• Overcomes dense Cholesky factors in sparse precision matrices
• Develops efficient conjugate gradient methods for large GLMMs

Innovation

• Conjugate gradient method for high-dimensional GLMMs
• Linear scaling with parameters and observations
• Sparse precision matrix spectral analysis