Causal ASCEND: Scalable Two-tier Causal Discovery on High Dimensional Multi-omics Data

Learning causal relations from observational data is challenging because the graph search space grows super-exponentially with the number of variables. Ordering-based methods reduce this space by first identifying the topological ordering, whereas continuous optimization methods explore most likely regions of the space by casting DAG learning as a differentiable objective with an acyclicity constraint. Despite their conceptual appeal, both paradigms face significant scalability limitations in high-dimensional settings, restricting their practical applicability. In this work, we introduce a new formulation for linear causal discovery that tightly integrates these two paradigms to achieve substantial gains in scalability without sacrificing accuracy. Our approach, TriOpt, decomposes the problem into two efficient stages. First, it recovers the topological ordering by exploiting the Sherman-Morrison rank-1 downdate together with the additive structure of linear kernels, enabling fast and scalable ordering estimation. Second, given this ordering, we reformulate structure learning as a convex continuous optimization problem that entirely avoids the need for enforcing costly acyclicity constraints. We theoretically show that, under the true ordering, TriOpt exactly recovers the underlying linear DAG. Empirically, across synthetic, semi-synthetic, and real-world datasets, TriOpt achieves orders-of-magnitude speedups over state-of-the-art linear causal discovery methods in high-dimensional regimes, while maintaining comparable or superior accuracy.

From climate teleconnections to gene regulation, modern time-series datasets encompass tens or hundreds of interacting variables, making causal discovery increasingly challenging. Constraint-based methods offer statistical rigor but their nonlinear CI tests are infeasible at scale, while score-based alternatives avoid CI testing but require arbitrary thresholds to binarize continuous edge scores. We propose GRACE ($\textbf{G}$ated $\textbf{R}$efinement for $\textbf{A}$ccurate $\textbf{C}$ausal $\textbf{E}$dge discovery), which refines constraint-based discovery using Hard Concrete gates with $L_0$ regularization: each candidate edge has an independent gate whose values concentrate near 0 or 1, yielding a clean bimodal separation that makes the binary decision robust, unlike the narrow, overlapping score distributions produced by $L_1$ and attention-based methods. A fast linear CI skeleton provides high-recall candidates; a single gated model then prunes false positives by learning which edges genuinely improve prediction, with automatic regularization adapted to problem dimensions and skeleton density. Systematic experiments on synthetic benchmarks, spanning diverse graph topologies (scale-free, Erdős-R'enyi, small-world) and dimensionalities up to $d=100$, show that GRACE substantially improves F1 over its base CI method while maintaining high precision, and outperforms attention-based and score-based alternatives. GRACE matches or exceeds expensive nonlinear CI tests at a fraction of the cost ($75\times$ faster). On a real-world river flow dataset, where rainfall confounders, variable propagation lags, and distributional shifts violate standard assumptions, a temporal bootstrap variant of GRACE recovers 9 of 11 causal edges along the Elbe River with only 1 false positive ($F_1 = 0.86$, AUROC${} = 0.99$), reducing the skeleton's 106 false positives by 99%.

Learning causal graphs from interventional data is a challenging problem with broad applications. In molecular biology, for example, a central goal is to uncover gene regulatory networks from large-scale perturbation data. An ideal algorithm for this task should scale to thousands of nodes, incorporate interventions even when their targets are unknown, quantify uncertainty, and provide identifiability guarantees. However, existing approaches---e.g. approaches using score-based optimization or approximate Bayesian inference---often fail to meet all of these criteria. To address these limitations, we develop Amortized Bayesian Causal Discovery of Extended Factor Graphs (ABCDEFG). Our method guarantees exact acyclicity, scales to graphs with thousands of nodes, and naturally handles interventions even when their targets are unknown. Additionally, ABCDEFG estimates a posterior distribution whose maximum a posteriori estimate provably identifies the true causal graph up to an equivalence class. On simulated datasets, ABCDEFG achieves state-of-the-art accuracy, producing a well-calibrated posterior distribution while outperforming previous score-based and approximate Bayesian methods. Applied to large-scale single-cell perturbation data, ABCDEFG identifies both established and novel gene targets of growth factors.