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RECLAIM: Cyclic Causal Discovery Amid Measurement Noise

Muralikrishnna G. Sethuraman, Faramarz Fekri

Abstract

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.

RECLAIM: Cyclic Causal Discovery Amid Measurement Noise

Abstract

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.
Paper Structure (51 sections, 11 theorems, 67 equations, 9 figures, 1 table, 1 algorithm)

This paper contains 51 sections, 11 theorems, 67 equations, 9 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related Works
  3. Cyclic causal discovery.
  4. Causal discovery from indirect measurements.
  5. Contributions
  6. Organization.
  7. Problem Setup
  8. Structural Causal Model
  9. Interventions.
  10. Measurement System
  11. RECLAIM Causal Discovery Framework
  12. Score Function
  13. Optimizing The Score Via Penalized Expectation-Maximization
  14. Estimation of Measurement System Parameters
  15. Gaussian Additive Noise.
  16. ...and 36 more sections

Key Result

Proposition 3

Let $\mathbf{A} \in \mathbb{R}^{p\times d}$ be the measurement matrix with $\rank(\mathbf{A}) = d$, let $\mathbold{a}_i$ denote the $i$-th column of $\mathbf{A}$, and let $\mathbf{A}_{-i}$ denote the measurement matrix excluding the $i$-th column. Then, there exists a vector $\mathbold{t} \in \mathb

Figures (9)

  • Figure 1: (a) Illustration of a causal graph $\mathcal{G}_m$ encoding both the system variables $\mathcal{X}$ and the measured variables $\mathcal{Y}$ for a linear measurement system. (b) Mutilated graph, $\text{do}(I)(\mathcal{G}_m)$, resulting from a surgical intervention on $X_2$.
  • Figure 2: Performance comparison with varying minimum noise standard deviation ($\sigma$) (top), and varying number of latent nodes ($d$) (bottom) for Gaussian additive noise system. Shaded regions show $\pm 1$ standard deviation over 10 trials.
  • Figure 3: Performance comparison with varying number of measurements ($p$) (top), and varying minimum noise standard deviation ($\sigma$) (bottom) for Linear measurement system. Shaded regions show $\pm 1$ standard deviation over 10 trials.
  • Figure 4: Estimated graph learnt from sachs_causal_2005 dataset
  • Figure 5: Illustration of state evolution for different choice of $a$ and $b$ for the system defined in \ref{['example:no-fixed-point']}. (Left) Choosing $a = 0.7$ and $b=-0.8$ results in a stable system and the system reaches a fixed point. (Right) Choosing $a = 1.2$ and $b = 2.3$ results in an unstable system and the trajectory diverges.
  • ...and 4 more figures

Theorems & Definitions (24)

  • Proposition 3
  • Theorem 4
  • Theorem 5
  • proof : Proof (sketch)
  • Example A.1
  • Definition A.2: Contractive function
  • proof
  • Lemma C.1
  • proof : \ref{['theorem:full-rank-possible']}
  • Definition C.2: $\sigma$-separation
  • ...and 14 more