Enhancing Graphical Lasso: A Robust Scheme for Non-Stationary Mean Data
Journal:
arXiv
Published Date:
Mar 25, 2025
Abstract
This work addresses the problem of graph learning from data following a
Gaussian Graphical Model (GGM) with a time-varying mean. Graphical Lasso (GL),
the standard method for estimating sparse precision matrices, assumes that the
observed data follows a zero-mean Gaussian distribution. However, this
assumption is often violated in real-world scenarios where the mean evolves
over time due to external influences, trends, or regime shifts. When the mean
is not properly accounted for, applying GL directly can lead to estimating a
biased precision matrix, hence hindering the graph learning task. To overcome
this limitation, we propose Graphical Lasso with Adaptive Targeted Adaptive
Importance Sampling (GL-ATAIS), an iterative method that jointly estimates the
time-varying mean and the precision matrix. Our approach integrates Bayesian
inference with frequentist estimation, leveraging importance sampling to obtain
an estimate of the mean while using a regularized maximum likelihood estimator
to infer the precision matrix. By iteratively refining both estimates, GL-ATAIS
mitigates the bias introduced by time-varying means, leading to more accurate
graph recovery. Our numerical evaluation demonstrates the impact of properly
accounting for time-dependent means and highlights the advantages of GL-ATAIS
over standard GL in recovering the true graph structure.
