Publications

Stochastic differential equations (SDEs) are widely used to model mechanistic systems evolving under random perturbations. For linear SDEs, the system's generator determines the post-intervention distributions, making its identifiability essential for causal inference. However, existing identifiability results assume the system is fully observed, whereas real systems are routinely subject to unmeasured confounders. To address this limitation, we study linear SDEs whose observed component is driven by a latent linear differential equation, considering both deterministic and stochastic confounding regimes. We characterize identifiable invariants comprising the observed drift, the diffusion Gram matrix, a family of mean-channel vectors, and (under stochastic confounding) a family of latent-noise matrices. We establish that identifiability holds if and only if a specific identifiability matrix constructed from Krylov subspaces of the identifiable invariants has full rank, which recovers the fully observed case as a special case. We validate these theoretical results through synthetic experiments that corroborate our findings.

Inferring causal relationships among variables from data is a central problem with applications across many domains. However, observational data alone generally identifies only an equivalence class rather than the exact causal graph. This challenge is more pronounced in systems with feedback loops, where observational data may fail to identify even the graph skeleton. Interventional data can resolve these ambiguities, but existing approaches often require designs covering all nodes, which can be costly in scientific applications. Moreover, active intervention selection has been studied primarily in acyclic settings. We propose PRISM, a parent-revealing active intervention selection framework for causal discovery in linear Gaussian cyclic structural equation models. PRISM combines a differentiable likelihood-based graph learner with two complementary criteria: an entropy-based criterion targeting nodes with uncertain incoming edges, and an equivalence-class reduction criterion selecting an intervention that maximally shrinks the remaining candidate graph class. We prove that a single-node hard intervention identifies the intervened node's parent vector. For any fixed candidate class, we further prove that the reduction objective is adaptive monotone and adaptive submodular, yielding the standard near-optimality guarantee for greedy intervention selection. Experiments on synthetic cyclic SEMs demonstrate near-perfect graph recovery with substantially fewer interventions than existing baselines.

This paper considers learning of the graphical structure of a $p$-dimensional random vector $\mathbf{X} \in \mathbb{R}^p$ using both parametric and non-parametric methods. Unlike the previous works which observe $\boldsymbol{x}$ directly, we consider the indirect observation scenario in which samples $\boldsymbol{y}$ are collected via a sensing matrix $\mathbf{A} \in \mathbb{R}^{d\times p}$, and corrupted with some additive noise $\mathbf{w}$, i.e, $\mathbf{Y} = \mathbf{A}\mathbf{X} + \mathbf{W}$. For the parametric method, we assume $\mathbf{X}$ to be Gaussian, i.e., $\boldsymbol{x} \in \mathbb{R}^p\sim \mathcal{N}\left(\mathbf{\mu}, \mathbf{\Sigma}\right)$, $\mathbf{\mu} \in \mathbb{R}^p$, and $\mathbf{\Sigma} \in \mathbb{R}^{p\times p}$. For the first time, we show that the correct graphical structure can be correctly recovered under the indefinite sensing system ($d < p$) using insufficient samples ($n < p$). In particular, we show that for the exact recovery, we require dimension $d = \Omega(p^{0.8})$ and sample number $n = \Omega(p^{0.8}\log^3 p)$. For the nonparametric method, we assume a nonparanormal distribution for $\mathbf{X}$ rather than Gaussian. Under mild conditions, we show that our graph-structure estimator can obtain the correct structure. We derive the minimum sample number $n$ and dimension $d$ as $n\gtrsim (\textup{deg})^4 \log^4 n$ and $d \gtrsim p + (\text{deg}\cdot\log(d-p))^{\beta/4})$, respectively, where $\textup{deg}$ is the maximum Markov blanket in the graphical model and $\beta > 0$ is some fixed positive constant. Additionally, we obtain a non-asymptotic uniform bound on the estimation error of the CDF of $\mathbf{X}$ from indirect observations with inexact knowledge of the noise distribution. To the best of our knowledge, this bound is derived for the first time and may serve as an independent interest. Numerical experiments on both real-world and synthetic data are provided confirm the theoretical results.

Uncovering causal relationships is a fundamental problem across science and engineering. However, most existing causal discovery methods assume acyclicity and direct access to the system variables—assumptions that fail to hold in many real-world settings. For instance, in genomics, cyclic regulatory networks are common, and measurements are often corrupted by instrumental noise. To address these challenges, we propose RECLAIM, a causal discovery framework that natively handles both cycles and measurement noise. RECLAIM learns the causal graph structure by maximizing the likelihood of the observed measurements via expectation-maximization (EM), using residual normalizing flows for tractable likelihood computation. We consider two measurement models: (i) Gaussian additive noise, and (ii) a linear measurement system with additive Gaussian noise. We provide theoretical consistency guarantees for both the settings. Experiments on synthetic data and real-world protein signaling datasets demonstrate the efficacy of the proposed method.

Learning causal relationships between variables from data is a fundamental research area with many applications across disciplines. Most of the existing causal discovery algorithms rely on the assumptions that (i) the underlying system is acyclic, (ii) the exogenous noise variables are Gaussian, and (iii) that the intervention targets for the data generating experiments are known. While these assumptions simplify the analysis, they are violated in real-life systems. Most existing methods that address these issues either assume the underlying model is linear or are constrained to operate in limited interventional settings. To that end, we propose SCOUT, a novel causal discovery framework to learn nonlinear causal cyclic relationships from soft interventional data with unknown targets. Our main approach maximizes the data log-likelihood to recover the graph structure, using two normalizing-flow architectures—contractive residual flows and neural spline flows. By conducting experiments on synthetic and real-world data, we show that SCOUT outperforms state-of-the-art methods in both causal graph and unknown target recovery across various interventional and noise settings.

Causal discovery in real-world systems, such as biological networks, is often complicated by feedback loops and incomplete data. Standard algorithms, which assume acyclic structures or fully observed data, struggle with these challenges. To address this gap, we propose MissNODAG, a differentiable framework for learning both the underlying cyclic causal graph and the missingness mechanism from partially observed data, including data missing not at random. Our framework integrates an additive noise model with an expectation-maximization procedure, alternating between imputing missing values and optimizing the observed data likelihood, to uncover both the cyclic structures and the missingness mechanism. We establish consistency guarantees under exact maximization of the score function in the large sample setting. Finally, we demonstrate the effectiveness of MissNODAG through synthetic experiments and an application to real-world gene perturbation data.

Understanding causal relationships between variables is fundamental across scientific disciplines. Most causal discovery algorithms rely on two key assumptions: (i) all variables are observed, and (ii) the underlying causal graph is acyclic. While these assumptions simplify theoretical analysis, they are often violated in real-world systems, such as biological networks. Existing methods that account for confounders either assume linearity or struggle with scalability. To address these limitations, we propose DCCD-CONF, a novel framework for differentiable learning of nonlinear cyclic causal graphs in the presence of unmeasured confounders using interventional data. Our approach alternates between optimizing the graph structure and estimating the confounder distribution by maximizing the log-likelihood of the data. Through experiments on synthetic data and real-world gene perturbation datasets, we show that DCCD-CONF outperforms state-of-the-art methods in both causal graph recovery and confounder identification. Additionally, we provide consistency guarantees for our framework, reinforcing its theoretical soundness.

Monitoring the levels of biomarkers for diagnostic applications has significant potential for impacts on patient care, but the measurement of all relevant biomarkers for a given set of conditions is often too expensive or unwieldy to be feasible at scale. Here, we propose a novel computational method for detecting changes in the levels of multiple target molecules from a complex sample via a small, cost-effective group of biosensors. We use the framework of density evolution (DE), a technique commonly used in the design of linear error-correcting codes for transmission over noisy channels, to develop an approach for localizing changes to a small subset of input signals based on a few simple output signals. As a biologically relevant testbed, we sought to detect the changes in the levels of multiple different microRNAs (miRNAs), which are nucleic acid molecules that are being increasingly studied and used as biomarkers. We accomplished this via the use of a class of molecules called "toehold switches" to create biosensors each capable of detecting multiple different miRNA sequences via a single output, with an overlap in sensitivity patterns between the different biosensors. A small number of these sensors were then used for inference of miRNA profiles. We demonstrate the potential utility of our approach with real data. Experimental results indicate the promising outcomes regarding the effectiveness of our method in detecting changes in miRNA concentrations.

Learning causal relationships between variables is a well-studied problem in statistics, with many important applications in science. However, modeling real-world systems remain challenging, as most existing algorithms assume that the underlying causal graph is acyclic. While this is a convenient framework for developing theoretical developments about causal reasoning and inference, the underlying modeling assumption is likely to be violated in real systems, because feedback loops are common (e.g., in biological systems). Although a few methods search for cyclic causal models, they usually rely on some form of linearity, which is also limiting, or lack a clear underlying probabilistic model. In this work, we propose a novel framework for learning nonlinear cyclic causal graphical models from interventional data, called NODAGS-Flow. We perform inference via direct likelihood optimization, employing techniques from residual normalizing flows for likelihood estimation. Through synthetic experiments and an application to single-cell high-content perturbation screening data, we show significant performance improvements with our approach compared to state-of-the-art methods with respect to structure recovery and predictive performance.

In this paper, we propose a general framework for designing sensing matrix $\boldsymbol{A} \in \mathbb{R}^{d\times p}$, for estimation of sparse covariance matrix from compressed measurements of the form $\boldsymbol{y} = \boldsymbol{A}\boldsymbol{x} + \boldsymbol{n}$, where $\boldsymbol{y}, \boldsymbol{n} \in \mathbb{R}^d$, and $\boldsymbol{x} \in \mathbb{R}^p$. By viewing covariance recovery as inference over factor graphs via message passing algorithm, ideas from coding theory, such as Density Evolution (DE), are leveraged to construct a framework for the design of the sensing matrix. The proposed framework can handle both (1) regular sensing, i.e., equal importance is given to all entries of the covariance, and (2) preferential sensing, i.e., higher importance is given to a part of the covariance matrix. Through experiments, we show that the sensing matrix designed via density evolution can match the state-of-the-art for covariance recovery in the regular sensing paradigm and attain improved performance in the preferential sensing regime. Additionally, we study the feasibility of causal graph structure recovery using the estimated covariance matrix obtained from the compressed measurements.