Abstract

Although joint regularization has become a widely used tool for structured multi-graph es

timation, 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.

Key words and phrases: Gaussian graphical models, High-dimensional statistics, Joint regularization, Multi-graph models, Simultaneous inference

Information

Preprint No.SS-2026-0078
Manuscript IDSS-2026-0078
Complete AuthorsQiao Zheng, Gefei Li, Yibo Yan, Riquan Zhang
Corresponding AuthorsRiquan Zhang
Emailszhangriquan@163.com

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Acknowledgments

This research of Riquan Zhang is supported by the National Natural Science Foundation of

China (No. 12371272 and 12531013). The research of Yibo Yan is supported by the the

National Natural Science Foundation of China (No. 12401390).

Supplementary Materials

The online Supplementary Material contains the related algorithms, detailed proofs, additional simulation results and a real-data example with p = 100.

Supplementary materials are available for download.