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Learning Identifiable Factorized Causal Representations of Cellular Responses

Haiyi Mao, Romain Lopez, Kai Liu, Jan-Christian Hütter, David Richmond, Panayiotis V. Benos, Lin Qiu

TL;DR

This work addresses how single-cell perturbation responses depend on biological context by proposing Factorized Causal Representation (FCR), an identifiable deep generative model that decomposes cellular states into covariate-specific ($\mathbf{z}_x$), interaction-specific ($\mathbf{z}_{tx}$), and treatment-specific ($\mathbf{z}_t$) latent blocks. Building on nonlinear ICA theory, it provides identifiability guarantees for $\mathbf{z}_{tx}$ and block-wise identifiability for $\mathbf{z}_x$ and $\mathbf{z}_t$ under rich experimental variability. The methodology combines variational inference with causal structure regularization and permutation discriminators to enforce the desired independence properties, and it validates on four real single-cell datasets, showing improved clustering and conditional independence testing results, as well as competitive single-cell response predictions. This framework advances precision medicine by enabling interpretable, interaction-aware representations of cellular responses across diverse biological contexts.

Abstract

The study of cells and their responses to genetic or chemical perturbations promises to accelerate the discovery of therapeutic targets. However, designing adequate and insightful models for such data is difficult because the response of a cell to perturbations essentially depends on its biological context (e.g., genetic background or cell type). For example, while discovering therapeutic targets, one may want to enrich for drugs that specifically target a certain cell type. This challenge emphasizes the need for methods that explicitly take into account potential interactions between drugs and contexts. Towards this goal, we propose a novel Factorized Causal Representation (FCR) learning method that reveals causal structure in single-cell perturbation data from several cell lines. Based on the framework of identifiable deep generative models, FCR learns multiple cellular representations that are disentangled, comprised of covariate-specific ($\mathbf{z}_x$), treatment-specific ($\mathbf{z}_{t}$), and interaction-specific ($\mathbf{z}_{tx}$) blocks. Based on recent advances in non-linear ICA theory, we prove the component-wise identifiability of $\mathbf{z}_{tx}$ and block-wise identifiability of $\mathbf{z}_t$ and $\mathbf{z}_x$. Then, we present our implementation of FCR, and empirically demonstrate that it outperforms state-of-the-art baselines in various tasks across four single-cell datasets.

Learning Identifiable Factorized Causal Representations of Cellular Responses

TL;DR

This work addresses how single-cell perturbation responses depend on biological context by proposing Factorized Causal Representation (FCR), an identifiable deep generative model that decomposes cellular states into covariate-specific (), interaction-specific (), and treatment-specific () latent blocks. Building on nonlinear ICA theory, it provides identifiability guarantees for and block-wise identifiability for and under rich experimental variability. The methodology combines variational inference with causal structure regularization and permutation discriminators to enforce the desired independence properties, and it validates on four real single-cell datasets, showing improved clustering and conditional independence testing results, as well as competitive single-cell response predictions. This framework advances precision medicine by enabling interpretable, interaction-aware representations of cellular responses across diverse biological contexts.

Abstract

The study of cells and their responses to genetic or chemical perturbations promises to accelerate the discovery of therapeutic targets. However, designing adequate and insightful models for such data is difficult because the response of a cell to perturbations essentially depends on its biological context (e.g., genetic background or cell type). For example, while discovering therapeutic targets, one may want to enrich for drugs that specifically target a certain cell type. This challenge emphasizes the need for methods that explicitly take into account potential interactions between drugs and contexts. Towards this goal, we propose a novel Factorized Causal Representation (FCR) learning method that reveals causal structure in single-cell perturbation data from several cell lines. Based on the framework of identifiable deep generative models, FCR learns multiple cellular representations that are disentangled, comprised of covariate-specific (), treatment-specific (), and interaction-specific () blocks. Based on recent advances in non-linear ICA theory, we prove the component-wise identifiability of and block-wise identifiability of and . Then, we present our implementation of FCR, and empirically demonstrate that it outperforms state-of-the-art baselines in various tasks across four single-cell datasets.

Paper Structure

This paper contains 64 sections, 3 theorems, 75 equations, 9 figures, 9 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related Work
  3. Learning Representations of Cellular Responses
  4. Identifiable non-linear Independent Component Analysis models
  5. Preliminaries
  6. Generative Model
  7. Model Identifiability
  8. Identifiability Results
  9. Component-wise identifiability of z_tx
  10. Block-wise identifiability of z_x and z_t
  11. Methodology
  12. Model Specification and Variational Inference
  13. Model Specification
  14. Variational Inference
  15. Causal Structure Regularization
  16. ...and 49 more sections

Key Result

Theorem 4.4

Let us first define $\mathbf{v}(\mathbf{z}_{tx}, \mathbf{t}, \mathbf{x})$ as the vector: where $q_{i}$ denotes the logarithm of probability density $p_{\mathbf{z}_i \mid \mathbf{t}, \mathbf{x}}$ for component $\mathbf{z}_i$. If in addition to the assumptions ass11, ass12, ass13, we assume that the $2n_{tx}$ vectors are linearly independent then $\mathbf{z}_{tx}$ is component-wise identifiable.

Figures (9)

  • Figure 1: Data generating process: shaded nodes denote observed variables. Empty nodes denote latent variables.
  • Figure 2: The illustration of FCR models. (a) is the component for $p(\mathbf{z}_x \mid\mathbf{x}), p(\mathbf{z}_t \mid \mathbf{t})$ and $p(\mathbf{z}_{tx} \mid\mathbf{t}, \mathbf{x})$ (b) is the component to estimate $q(\mathbf{z}_x \mid\mathbf{x}, \mathbf{y}), q(\mathbf{z}_t \mid\mathbf{t}, \mathbf{y})$ and $q(\mathbf{z}_{t,x} \mid \mathbf{t}, \mathbf{x}, \mathbf{y})$. Note that $\mathbf{0}$ indicates $\mathbf{t}=\mathbf{0}$ representing the control samples. (c) computational diagrams to estimate the Kullback-Leibler divergences, causal structure regularization and permutation discriminators.
  • Figure 3: Clustering results for the sciPlex dataset. Normalized Mutual Information (NMI) values for clustering based on: (a) covariates $\mathbf{x}$; (b) combined covariates and treatments $[\mathbf{x}; \mathbf{t}]$; (c) treatments $\mathbf{t}$.
  • Figure 4: Statistical Conditional Independence Testing Results (a) p-values for the conditional independence test of $\mathbf{z}_x \perp \!\!\! \perp \mathbf{t} \mid \mathbf{x}$. The red dashed line indicates the 0.05 level. (b) p-values for the conditional independence test of $\mathbf{z}_t \perp \!\!\! \perp \mathbf{x} \mid \mathbf{t}$. (c) p-values for the conditional independence tests $\mathbf{z}_x \perp \!\!\! \perp \mathbf{z}_{tx} \mid \mathbf{x}$ and $\mathbf{z}_t \perp \!\!\! \perp \mathbf{z}_{tx} \mid \mathbf{t}$. (d) HSIC values for assessment of marginal independence of $\mathbf{z}_x$ with $\mathbf{x}$, $\mathbf{t}$ and random numbers ($\mathbf{R}$); as well as $\mathbf{z}_t$ with $\mathbf{x}$, $\mathbf{t}$ and random numbers ($\mathbf{R}$).
  • Figure 5: The clustering results. (a) For the multiPlex-Tram dataset, the first column shows the NMI value for clustering on $\mathbf{x}$, the second column shows the NMI value for clustering on $\mathbf{xt}$, and the third column shows the NMI value for clustering on $\mathbf{t}$.(b) For the multiPlex-5 dataset, the first column presents the NMI value for clustering on $\mathbf{x}$, the second column shows the NMI value for clustering on $\mathbf{xt}$, and the third column shows the NMI value for clustering on $\mathbf{t}$. (c) For the multiPlex-9 dataset, the first column displays the NMI value for clustering on $\mathbf{x}$, the second column shows the NMI value for clustering on $\mathbf{xt}$, and the third column shows the NMI value for clustering on $\mathbf{t}$.
  • ...and 4 more figures

Theorems & Definitions (7)

  • Definition 3.1: Component-wise Identifiability
  • Definition 3.2: Block-wise Identifiability
  • Theorem 4.4
  • Theorem 4.5
  • Proposition 5.1
  • proof
  • proof