Toward the Identifiability of Comparative Deep Generative Models

Deep Generative Models (DGMs) are versatile tools for learning data representations while adequately incorporating domain knowledge such as the specification of conditional probability distributions. Recently proposed DGMs tackle the important task of comparing data sets from different sources. One such example is the setting of contrastive analysis that focuses on describing patterns that are enriched in a target data set compared to a background data set. The practical deployment of those models often assumes that DGMs naturally infer interpretable and modular latent representations, which is known to be an issue in practice. Consequently, existing methods often rely on ad-hoc regularization schemes, although without any theoretical grounding. Here, we propose a theory of identifiability for comparative DGMs by extending recent advances in the field of non-linear independent component analysis. We show that, while these models lack identifiability across a general class of mixing functions, they surprisingly become identifiable when the mixing function is piece-wise affine (e.g., parameterized by a ReLU neural network). We also investigate the impact of model misspecification, and empirically show that previously proposed regularization techniques for fitting comparative DGMs help with identifiability when the number of latent variables is not known in advance. Finally, we introduce a novel methodology for fitting comparative DGMs that improves the treatment of multiple data sources via multi-objective optimization and that helps adjust the hyperparameter for the regularization in an interpretable manner, using constrained optimization. We empirically validate our theory and new methodology using simulated data as well as a recent data set of genetic perturbations in cells profiled via single-cell RNA sequencing.

Learning Causal Representations of Single Cells via Sparse Mechanism Shift Modeling

Latent variable models such as the Variational Auto-Encoder (VAE) have become a go-to tool for analyzing biological data, especially in the field of single-cell genomics. One remaining challenge is the interpretability of latent variables as biological processes that define a cell's identity. Outside of biological applications, this problem is commonly referred to as learning disentangled representations. Although several disentanglement-promoting variants of the VAE were introduced, and applied to single-cell genomics data, this task has been shown to be infeasible from independent and identically distributed measurements, without additional structure. Instead, recent methods propose to leverage non-stationary data, as well as the sparse mechanism shift assumption in order to learn disentangled representations with a causal semantic. Here, we extend the application of these methodological advances to the analysis of single-cell genomics data with genetic or chemical perturbations. More precisely, we propose a deep generative model of single-cell gene expression data for which each perturbation is treated as a stochastic intervention targeting an unknown, but sparse, subset of latent variables. We benchmark these methods on simulated single-cell data to evaluate their performance at latent units recovery, causal target identification and out-of-domain generalization. Finally, we apply those approaches to two real-world large-scale gene perturbation data sets and find that models that exploit the sparse mechanism shift hypothesis surpass contemporary methods on a transfer learning task. We implement our new model and benchmarks using the scvi-tools library, and release it as open-source software at https://github.com/Genentech/sVAE.

Large-Scale Differentiable Causal Discovery of Factor Graphs

A common theme in causal inference is learning causal relationships between observed variables, also known as causal discovery. This is usually a daunting task, given the large number of candidate causal graphs and the combinatorial nature of the search space. Perhaps for this reason, most research has so far focused on relatively small causal graphs, with up to hundreds of nodes. However, recent advances in fields like biology enable generating experimental data sets with thousands of interventions followed by rich profiling of thousands of variables, raising the opportunity and urgent need for large causal graph models. Here, we introduce the notion of factor directed acyclic graphs (f-DAGs) as a way to restrict the search space to non-linear low-rank causal interaction models. Combining this novel structural assumption with recent advances that bridge the gap between causal discovery and continuous optimization, we achieve causal discovery on thousands of variables. Additionally, as a model for the impact of statistical noise on this estimation procedure, we study a model of edge perturbations of the f-DAG skeleton based on random graphs and quantify the effect of such perturbations on the f-DAG rank. This theoretical analysis suggests that the set of candidate f-DAGs is much smaller than the whole DAG space and thus more statistically robust in the high-dimensional regime where the underlying skeleton is hard to assess. We propose Differentiable Causal Discovery of Factor Graphs (DCD-FG), a scalable implementation of f-DAG constrained causal discovery for high-dimensional interventional data. DCD-FG uses a Gaussian non-linear low-rank structural equation model and shows significant improvements compared to state-of-the-art methods in both simulations as well as a recent large-scale single-cell RNA sequencing data set with hundreds of genetic interventions.

Inference of cell dynamics on perturbation data using adjoint sensitivity

Data-driven dynamic models of cell biology can be used to predict cell response to unseen perturbations. Recent work (CellBox) had demonstrated the derivation of interpretable models with explicit interaction terms, in which the parameters were optimized using machine learning techniques. While the previous work was tested only in a single biological setting, this work aims to extend the range of applicability of this model inference approach to a diversity of biological systems. Here we adapted CellBox in Julia differential programming and augmented the method with adjoint algorithms, which has recently been used in the context of neural ODEs. We trained the models using simulated data from both abstract and biology-inspired networks, which afford the ability to evaluate the recovery of the ground truth network structure. The resulting accuracy of prediction by these models is high both in terms of low error against data and excellent agreement with the network structure used for the simulated training data. While there is no analogous ground truth for real life biological systems, this work demonstrates the ability to construct and parameterize a considerable diversity of network models with high predictive ability. The expectation is that this kind of procedure can be used on real perturbation-response data to derive models applicable to diverse biological systems.

Reconstructing probabilistic trees of cellular differentiation from single-cell RNA-seq data

Until recently, transcriptomics was limited to bulk RNA sequencing, obscuring the underlying expression patterns of individual cells in favor of a global average. Thanks to technological advances, we can now profile gene expression across thousands or millions of individual cells in parallel. This new type of data has led to the intriguing discovery that individual cell profiles can reflect the imprint of time or dynamic processes. However, synthesizing this information to reconstruct dynamic biological phenomena from data that are noisy, heterogenous, and sparse---and from processes that may unfold asynchronously---poses a complex computational and statistical challenge. Here, we develop a full generative model for probabilistically reconstructing trees of cellular differentiation from single-cell RNA-seq data. Specifically, we extend the framework of the classical Dirichlet diffusion tree to simultaneously infer branch topology and latent cell states along continuous trajectories over the full tree. In tandem, we construct a novel Markov chain Monte Carlo sampler that interleaves Metropolis-Hastings and message passing to leverage model structure for efficient inference. Finally, we demonstrate that these techniques can recover latent trajectories from simulated single-cell transcriptomes. While this work is motivated by cellular differentiation, we derive a tractable model that provides flexible densities for any data (coupled with an appropriate noise model) that arise from continuous evolution along a latent nonparametric tree.

Learning Module Networks

Methods for learning Bayesian network structure can discover dependency structure between observed variables, and have been shown to be useful in many applications. However, in domains that involve a large number of variables, the space of possible network structures is enormous, making it difficult, for both computational and statistical reasons, to identify a good model. In this paper, we consider a solution to this problem, suitable for domains where many variables have similar behavior. Our method is based on a new class of models, which we call module networks. A module network explicitly represents the notion of a module - a set of variables that have the same parents in the network and share the same conditional probability distribution. We define the semantics of module networks, and describe an algorithm that learns a module network from data. The algorithm learns both the partitioning of the variables into modules and the dependency structure between the variables. We evaluate our algorithm on synthetic data, and on real data in the domains of gene expression and the stock market. Our results show that module networks generalize better than Bayesian networks, and that the learned module network structure reveals regularities that are obscured in learned Bayesian networks.