Network-aware IV Regression for Causal Node Discovery and Estimation

Abstract: Estimating causal effects from high-dimensional, structured exposures is a fundamental challenge in modern applications ranging from neuroscience and finance to environmental science. While the literature has addressed high-dimensional instrumental variable (IV) regression, and separately leveraged graph structure in penalized regression, the integration of both, especially for causal support recovery in the presence of latent confounding, remains unexplored. In this work, we propose a novel two-stage regression framework that incorporates instrumental variables and graph-based regularization to uncover sparse causal effects among network-structured exposures. Our method accommodates both valid and partially invalid instruments, and encourages structural similarity among connected predictors through a graph-fused penalty. We establish non-asymptotic guarantees for estimation accuracy and causal variable selection, and demonstrate that our approach yields improved performance over existing methods that ignore network dependencies or invalid IVs. Applied to ADNI brain imaging and genetic data, our method identifies interpretable causal ROIs associated with cognitive outcomes, underscoring the utility of graph-assisted IV regression in neuroscience and beyond.

• The paper introduces IVGL, a two-stage instrumental-variable method that combines LASSO sparsity with graph-based coefficient fusion to identify and estimate causal effects among connected, high-dimensional exposures under latent confounding.
• Theoretical results provide non-asymptotic error bounds and support-recovery guarantees, while simulations show lower MSE and stronger discovery performance than network-agnostic IV-LASSO, including roughly 10^-4 versus 0.05–0.13 MSE when instruments are invalid.
• The IVGL-S extension handles sparse instrument violations and, in ADNI data, narrows 62 brain regions to eight interpretable medial and superior temporal candidates associated with cognitive performance and Alzheimer’s pathology.

This paper develops a two-stage instrumental variable (IV) regression framework that integrates graph-structured regularization over high-dimensional exposures, enabling causal node discovery and effect estimation when predictors are connected on a known network and latent confounding is present. The work addresses a gap at the intersection of two literatures: network-constrained penalized regression, which is purely associational, and high-dimensional IV methods such as sisVIVE (2604.24969). To the authors' knowledge, no prior method provides formal causal identification for node effects when the covariates themselves form a network.

Methodology

The proposed framework, IVGL (Instrumental Variable Graph LASSO), assumes each subject has an exposure vector $\mathbf{x}_i \in \mathbb{R}^p$ of network-valued covariates, a set of candidate instruments $\mathbf{Z}_i \in \mathbb{R}^q$, and an outcome $Y_i$. In the first stage, each exposure component is regressed on the instruments via LASSO, yielding predicted exposures $\hat{X} = Z\hat{\bm A}$. In the second stage, the outcome is regressed on $\hat{\mathbf{x}}_i$ under a combined penalty:

$\mathcal{P}_{\lambda_1,\lambda_2}(\bm\beta) = \lambda_1 \| \boldsymbol{\beta} \|_1 + \lambda_2 \sum_{(j,k) \in E} w_{jk} \left({\beta_j}/{\sqrt{d_j} - {\beta_k}/{\sqrt{d_k}\right)^2,$

where $G = (V,E)$ is the known predictor network. The $\ell_1$ term enforces sparsity while the Laplacian-based fusion term encourages similar coefficients among connected nodes, increasing power to detect clusters of related causal exposures rather than isolated signals.

To accommodate invalid instruments—e.g., horizontal pleiotropy in Mendelian randomization—the authors extend the framework to IVGL-S by building on sisVIVE (2604.24969). The outcome model includes direct instrument effects, $$Y_i = \mathbf{x}_i^\mathrm{T}\boldsymbol{\beta} + \mathbf{Z}_i^\mathrm{T}\boldsymbol{\alpha} + \varepsilon_i$$, and estimation proceeds by projecting onto the column space of $Z$, minimizing the projected residual norm jointly over $\mathbf{Z}_i \in \mathbb{R}^q$0 with an additional $\mathbf{Z}_i \in \mathbb{R}^q$1 penalty on $\mathbf{Z}_i \in \mathbb{R}^q$2 to induce sparsity among direct effects. This filters out outcome variation outside the span of the instruments, aligning conceptually with 2SLS while remaining robust to sparse exclusion-restriction violations.

Theoretical guarantees

The theory section establishes non-asymptotic results under sub-Gaussian tail assumptions, a compatibility condition on the true design, and a modified restricted strong convexity (RSC) condition. The modified RSC is notable: it explicitly accounts for curvature lost to first-stage estimation error (an $\mathbf{Z}_i \in \mathbb{R}^q$3 term) and curvature gained from the graph Laplacian, reducing to standard RSC when there is neither first-stage noise nor network penalty.

Theorem 1 gives an oracle-type $\mathbf{Z}_i \in \mathbb{R}^q$4 error bound for $\mathbf{Z}_i \in \mathbb{R}^q$5: $\mathbf{Z}_i \in \mathbb{R}^q$6, recovering the familiar $\mathbf{Z}_i \in \mathbb{R}^q$7 rate when $\mathbf{Z}_i \in \mathbb{R}^q$8. A corollary quantifies the benefit of the graph penalty: if the Laplacian contributes extra cone-restricted curvature ($\mathbf{Z}_i \in \mathbb{R}^q$9), then choosing $Y_i$0 yields a strictly tighter bound than the no-graph two-stage LASSO whenever $Y_i$1 and the graph-smoothness radius $Y_i$2 is finite. This formalizes the intuition that borrowing strength across edges improves estimation.

For support recovery, the paper introduces a network-aware irrepresentability condition, extending the classical condition of Zhao and Yu to account for Laplacian smoothing: inactive coefficients must not be representable by signed combinations of active ones after accounting for the graph regularizer. Combined with a beta-min condition whose threshold includes both $Y_i$3 and the Laplacian shrinkage term $Y_i$4, Theorem 2 proves selection and sign consistency of $Y_i$5.

For the invalid-instrument case, Theorem 3 shows IVGL-S achieves the same oracle rates as IVGL under a restricted isometry property (RIP) capturing the joint geometry of $Y_i$6, $Y_i$7, and their projections, provided $Y_i$8 and tuning parameters scale appropriately. A parallel corollary (in supplementary material) states conditions under which the graph constraint strictly tightens the bound relative to original sisVIVE.

Simulation evidence

Two simulation designs mimic neuroimaging-genetic data ($Y_i$9, $\hat{X} = Z\hat{\bm A}$0 ROIs, $\hat{X} = Z\hat{\bm A}$1 SNPs) with unmeasured confounding affecting both exposures and outcome. In Setup 1 (all IVs valid), IVGL consistently achieved smaller MSE and higher MCC than the network-ignoring baseline IV-LASSO across signal strengths $\hat{X} = Z\hat{\bm A}$2 and sparsity levels $\hat{X} = Z\hat{\bm A}$3; e.g., at $\hat{X} = Z\hat{\bm A}$4, $\hat{X} = Z\hat{\bm A}$5, MSE was 0.019 versus 0.048 for the baseline.

In Setup 2 (10 of 500 instruments invalid with direct effects of magnitude 5), the contrast is stark: IVGL-S MSEs were several orders of magnitude smaller than IVGL's—for instance, roughly $\hat{X} = Z\hat{\bm A}$6 versus $\hat{X} = Z\hat{\bm A}$7–$\hat{X} = Z\hat{\bm A}$8—with near-perfect support recovery by MCC, while IVGL degraded severely. The gap widened with weaker signals or greater sparsity, indicating that correction for invalid IVs is critical precisely when true causal effects are faint or localized. These results validate that the graph-fusion advantage survives the sisVIVE augmentation.

Application to ADNI

The framework is applied to baseline ADNI data (484 participants: 151 CN, 231 MCI, 102 AD), with MMSE score as a continuous cognitive endpoint, 62 DKT-atlas ROI features from T1-weighted MRI as exposures, and ~20,000 SNPs screened to 300 candidate instruments via sure independence screening. A three-stage pipeline—graph-constrained LASSO (associational), IVGL, and IVGL-S—yields progressively more conservative ROI sets: GL selects 17 regions bilaterally; IVGL refines this to 13; IVGL-S retains only 8, concentrated in medial and superior temporal structures including bilateral entorhinal cortex, insula, superior temporal, and transverse temporal gyri. The progressive dropout of frontal and parietal regions under IVGL-S suggests those associations were largely confounded, while the surviving medial temporal ROIs align with established early Alzheimer's pathology (Braak staging, entorhinal degeneration). Compared with the CNN-based DeepFEIVR approach, the selected regions here retain direct anatomical interpretability.

Limitations and open questions

Several caveats bear directly on the results. The theoretical guarantees rest on strong assumptions—sub-Gaussian tails, compatibility, the modified RSC, network irrepresentability, beta-min, and RIP conditions—whose empirical plausibility in real neuroimaging-genetic data is not verified. The real-data analysis is cross-sectional despite ADNI's longitudinal design, so the identified ROIs concern baseline cognition rather than progression rates; extending the framework to longitudinal or dynamic-graph settings remains open. The beta-min threshold grows with the Laplacian shrinkage term, meaning very weak but genuine effects adjacent to strong ones may be missed—a trade-off inherent to graph fusion that the paper does not quantify empirically. Finally, identifiability under IVGL-S requires the invalid instruments to be sparse, and the sensitivity of results to the degree of pleiotropy is not assessed.

Conclusion

The paper contributes a principled integration of graph-regularized penalization with high-dimensional IV regression, supported by non-asymptotic estimation and selection guarantees for both valid and partially invalid instruments, and demonstrates substantial empirical gains—particularly the orders-of-magnitude MSE improvement of IVGL-S under invalid IVs. Its application to ADNI produces anatomically coherent, interpretable causal ROI candidates centered on the medial temporal lobe, illustrating the utility of structured causal discovery in neuroimaging and related domains with network-structured exposures.