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Linear Causal Representation Learning from Unknown Multi-node Interventions

Burak Varıcı, Emre Acartürk, Karthikeyan Shanmugam, Ali Tajer

TL;DR

This paper tackles the identifiability problem in causal representation learning when interventional environments involve unknown multi-node perturbations (UMN). It develops a score-based framework that links changes in latent and observed score functions across environments to recover latent factors and their DAG structure under both soft and hard UMN interventions. The authors prove perfect identifiability for linear transformations with regular UMN hard interventions and identifiability up to ancestors for UMN soft interventions, providing constructive proofs and an accompanying UMNI-CRL algorithm with four stages. Simulations demonstrate strong latent-variable recovery and accurate graph estimation up to $n=8$ latent nodes, illustrating practical viability in settings with high-dimensional observations. The work opens avenues for extending to nonlinearity and refining the necessary diversity conditions on UMN interventions for identifiability in broader models.

Abstract

Despite the multifaceted recent advances in interventional causal representation learning (CRL), they primarily focus on the stylized assumption of single-node interventions. This assumption is not valid in a wide range of applications, and generally, the subset of nodes intervened in an interventional environment is fully unknown. This paper focuses on interventional CRL under unknown multi-node (UMN) interventional environments and establishes the first identifiability results for general latent causal models (parametric or nonparametric) under stochastic interventions (soft or hard) and linear transformation from the latent to observed space. Specifically, it is established that given sufficiently diverse interventional environments, (i) identifiability up to ancestors is possible using only soft interventions, and (ii) perfect identifiability is possible using hard interventions. Remarkably, these guarantees match the best-known results for more restrictive single-node interventions. Furthermore, CRL algorithms are also provided that achieve the identifiability guarantees. A central step in designing these algorithms is establishing the relationships between UMN interventional CRL and score functions associated with the statistical models of different interventional environments. Establishing these relationships also serves as constructive proof of the identifiability guarantees.

Linear Causal Representation Learning from Unknown Multi-node Interventions

TL;DR

This paper tackles the identifiability problem in causal representation learning when interventional environments involve unknown multi-node perturbations (UMN). It develops a score-based framework that links changes in latent and observed score functions across environments to recover latent factors and their DAG structure under both soft and hard UMN interventions. The authors prove perfect identifiability for linear transformations with regular UMN hard interventions and identifiability up to ancestors for UMN soft interventions, providing constructive proofs and an accompanying UMNI-CRL algorithm with four stages. Simulations demonstrate strong latent-variable recovery and accurate graph estimation up to latent nodes, illustrating practical viability in settings with high-dimensional observations. The work opens avenues for extending to nonlinearity and refining the necessary diversity conditions on UMN interventions for identifiability in broader models.

Abstract

Despite the multifaceted recent advances in interventional causal representation learning (CRL), they primarily focus on the stylized assumption of single-node interventions. This assumption is not valid in a wide range of applications, and generally, the subset of nodes intervened in an interventional environment is fully unknown. This paper focuses on interventional CRL under unknown multi-node (UMN) interventional environments and establishes the first identifiability results for general latent causal models (parametric or nonparametric) under stochastic interventions (soft or hard) and linear transformation from the latent to observed space. Specifically, it is established that given sufficiently diverse interventional environments, (i) identifiability up to ancestors is possible using only soft interventions, and (ii) perfect identifiability is possible using hard interventions. Remarkably, these guarantees match the best-known results for more restrictive single-node interventions. Furthermore, CRL algorithms are also provided that achieve the identifiability guarantees. A central step in designing these algorithms is establishing the relationships between UMN interventional CRL and score functions associated with the statistical models of different interventional environments. Establishing these relationships also serves as constructive proof of the identifiability guarantees.
Paper Structure (44 sections, 9 theorems, 127 equations, 1 figure, 2 tables, 1 algorithm)

This paper contains 44 sections, 9 theorems, 127 equations, 1 figure, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related literature
  3. CRL setting and preliminaries
  4. Latent causal model
  5. Unknown multi-node intervention models
  6. Identifiability criteria
  7. Identifiability under UMN interventions
  8. UMN interventional CRL algorithm
  9. Simulations
  10. Discussion
  11. Proofs
  12. Additional notations.
  13. Definition 2 (Intervention regularity)
  14. Proof of Lemma \ref{['lemma:causal-order']}
  15. Base case.
  16. ...and 29 more sections

Key Result

Theorem 1

Under Assumption assumption:full-rank-environments and a latent model with additive noise,

Figures (1)

  • Figure 1: Sensitivity analysis of UMNI-CRL algorithm for quadratic latent causal models. The results are for $n=5$ latent nodes and $d=20$ observed variables, $10^4$ samples, and for average of 100 runs. (a):${\rm SHD}(\mathcal{G}_{\rm tc}, \hat{\mathcal{G}})$ versus SNR (soft) and ${\rm SHD}(\mathcal{G}, \hat{\mathcal{G}})$ versus SNR (hard). (b): Incorrect mixing ratio $\ell_{\rm soft}$ versus SNR (soft) and $\ell_{\rm hard}$ versus SNR (hard).

Theorems & Definitions (17)

  • Definition 1: Identifiability
  • Definition 2: Intervention regularity
  • Theorem 1: Identifiability under UMN hard interventions
  • Theorem 2: Identifiability under UMN soft interventions
  • Lemma 1: varici2024score
  • Lemma 2
  • proof
  • Theorem 3
  • proof
  • Theorem 4
  • ...and 7 more