  Many causal systems such as biological processes in cells can only be
observed indirectly via measurements, such as gene expression. Causal
representation learning -- the task of correctly mapping low-level observations
to latent causal variables -- could advance scientific understanding by
enabling inference of latent variables such as pathway activation. In this
paper, we develop methods for inferring latent variables from multiple related
datasets (environments) and tasks. As a running example, we consider the task
of predicting a phenotype from gene expression, where we often collect data
from multiple cell types or organisms that are related in known ways. The key
insight is that the mapping from latent variables driven by gene expression to
the phenotype of interest changes sparsely across closely related environments.
To model sparse changes, we introduce Tree-Based Regularization (TBR), an
objective that minimizes both prediction error and regularizes closely related
environments to learn similar predictors. We prove that under assumptions about
the degree of sparse changes, TBR identifies the true latent variables up to
some simple transformations. We evaluate the theory empirically with both
simulations and ground-truth gene expression data. We find that TBR recovers
the latent causal variables better than related methods across these settings,
even under settings that violate some assumptions of the theory.
