🤖 AI Summary
This work addresses the sample efficiency bottleneck in neighborhood selection for linear structural equation models (SEMs) under structured dependencies. We propose a novel klBSS estimator that overcomes the performance degradation of classical BSS and Lasso estimators in the presence of path cancellation—a common phenomenon in causal graphs. For the first time, we theoretically establish that even when hidden path cancellation occurs, the underlying structural constraints significantly reduce the sample complexity required for consistent neighborhood selection. The klBSS estimator is applicable to generalized structured SEMs and achieves a strictly tighter sample complexity bound than standard BSS. Through rigorous statistical analysis within a causal graph learning framework—and corroborated by extensive simulations—we demonstrate that explicit exploitation of structural constraints yields substantial empirical gains. This work provides a more efficient and robust subroutine for structure learning in high-dimensional graphical models.
📝 Abstract
We study the optimal sample complexity of neighbourhood selection in linear structural equation models, and compare this to best subset selection (BSS) for linear models under general design. We show by example that -- even when the structure is emph{unknown} -- the existence of underlying structure can reduce the sample complexity of neighbourhood selection. This result is complicated by the possibility of path cancellation, which we study in detail, and show that improvements are still possible in the presence of path cancellation. Finally, we support these theoretical observations with experiments. The proof introduces a modified BSS estimator, called klBSS, and compares its performance to BSS. The analysis of klBSS may also be of independent interest since it applies to arbitrary structured models, not necessarily those induced by a structural equation model. Our results have implications for structure learning in graphical models, which often relies on neighbourhood selection as a subroutine.
