A Unified Debiasing Framework for Inference with Jointly Regularized Multi-Graph Estimators

Although joint regularization has become a widely used tool for structured multi-graph estimation, statistical inference for multi-graph models is challenging because of the nontrivial coupling effects induced by cross-group penalties.To address this issue, we develop a KKT-based unified debiasing framework for inference with jointly regularized multi-graph estimators and derive an analytic debiasing representation that removes the leading regularization-induced bias while accounting for cross-group coupling.The resulting debiasing representation also admits an identity-based interpretation through the precision-covariance relation, highlighting its applicability beyond likelihood-based constructions to a broad class of multi-graph estimators satisfying the required consistency conditions under standard high-dimensional sparsity regimes.We establish theoretical guarantees, including estimation consistency and entry-wise asymptotic normality, and develop inferential procedures that provide entry-wise confidence intervals and simultaneous tests with asymptotically valid control of the family-wise error rate (FWER) and the false discovery rate (FDR).We further apply the proposed framework to differential structure detection to identify structural changes across groups.Extensive simulation studies and a real-data analysis demonstrate the validity and effectiveness of the proposed debiasing framework for high-dimensional multi-graph inference.