Advances in sequencing technologies have enhanced the understanding of gene regulation in cells. In particular, Perturb-seq has enabled high-resolution profiling of the transcriptomic response to genetic perturbations at the single-cell level. This understanding has implications in functional genomics and potentially for identifying therapeutic targets. Various computational models have been developed to predict perturbational effects. While deep learning models excel at interpolating observed perturbational data, they tend to overfit in the lack of enough data and may not generalize well to unseen perturbations. In contrast, mechanistic models, such as linear causal models based on gene regulatory networks, hold greater potential for extrapolation, as they encapsulate regulatory information that can predict responses to unseen perturbations. However, their application has been limited to small studies due to overly simplistic assumptions, making them less effective in handling noisy, large-scale single-cell data. We propose a hybrid approach that combines a mechanistic causal model with variational deep learning, termed Single Cell Causal Variational Autoencoder (SCCVAE). The mechanistic model employs a learned regulatory network to represent perturbational changes as shift interventions that propagate through the learned network. SCCVAE integrates this mechanistic causal model into a variational autoencoder, generating rich, comprehensive transcriptomic responses. Our results indicate that SCCVAE exhibits superior performance over current state-of-the-art baselines for extrapolating to predict unseen perturbational responses. Additionally, for the observed perturbations, the latent space learned by SCCVAE allows for the identification of functional perturbation modules and simulation of single-gene knockdown experiments of varying penetrance, presenting a robust tool for interpreting and interpolating perturbational responses at the single-cell level.

A combinatorial intervention, consisting of multiple treatments applied to a single unit with potential interactive effects, has substantial applications in fields such as biomedicine, engineering, and beyond. Given $p$ possible treatments, conducting all possible $2^p$ combinatorial interventions can be laborious and quickly becomes infeasible as $p$ increases. Here we introduce the probabilistic factorial experimental design, formalized from how scientists perform lab experiments. In this framework, the experimenter selects a dosage for each possible treatment and applies it to a group of units. Each unit independently receives a random combination of treatments, sampled from a product Bernoulli distribution determined by the dosages. Additionally, the experimenter can carry out such experiments over multiple rounds, adapting the design in an active manner. We address the optimal experimental design problem within a novel intervention model that imposes bounded-degree interactions between treatments. In the passive setting, we provide a closed-form solution for the near-optimal design. Our results prove that a dosage of $\frac{1}{2}$ for each treatment is optimal up to a factor of $1+O(\frac{\ln(n)}{n})$ for estimating any $k$-way interaction model, regardless of $k$, and imply that $O\big(kp^{3k}\ln(p)\big)$ observations are required to accurately estimate this model. For the multi-round setting, we provide a near-optimal acquisition function that can be numerically optimized. We also explore several extensions of the design problem and finally validate our findings through simulations.

This paper presents a novel theoretical framework for understanding how diffusion models can learn disentangled representations with commonly used weak supervision such as partial labels and multiple views. Within this framework, we establish identifiability conditions for diffusion models to disentangle latent variable models with \emph{stochastic}, \emph{non-invertible} mixing processes. We also prove \emph{finite-sample global convergence} for diffusion models to disentangle independent subspace models. To validate our theory, we conduct extensive disentanglement experiments on subspace recovery in latent subspace Gaussian mixture models, image colorization, denoising, and voice conversion for speech classification. Our experiments show that training strategies inspired by our theory, such as style guidance regularization, consistently enhance disentanglement performance.

Understanding causal relationships between variables is a fundamental problem with broad impact in numerous scientific fields. While extensive research has been dedicated to \emph{learning} causal graphs from data, its complementary concept of \emph{testing} causal relationships has remained largely unexplored. While \emph{learning} involves the task of recovering the Markov equivalence class (MEC) of the underlying causal graph from observational data, the \emph{testing} counterpart addresses the following critical question: \emph{Given a specific MEC and observational data from some causal graph, can we determine if the data-generating causal graph belongs to the given MEC?} We explore constraint-based testing methods by establishing bounds on the required number of conditional independence tests. Our bounds are in terms of the size of the maximum undirected clique ($s$) of the given MEC. In the worst case, we show a lower bound of $\exp(\Omega(s))$ independence tests. We then give an algorithm that resolves the task with $\exp(O(s))$ tests, matching our lower bound. Compared to the \emph{learning} problem, where algorithms often use a number of independence tests that is exponential in the maximum in-degree, this shows that \emph{testing} is relatively easier. In particular, it requires exponentially less independence tests in graphs featuring high in-degrees and small clique sizes. Additionally, using the DAG associahedron, we provide a geometric interpretation of testing versus learning and discuss how our testing result can aid learning.

Many questions in science center around the fundamental problem of understanding causal relationships. However, most constraint-based causal discovery algorithms, including the well-celebrated PC algorithm, often incur an exponential number of conditional independence (CI) tests, posing limitations in various applications. Addressing this, our work focuses on characterizing what can be learned about the underlying causal graph with a reduced number of CI tests. We show that it is possible to a learn a coarser representation of the hidden causal graph with a polynomial number of tests. This coarser representation, named Causal Consistent Partition Graph (CCPG), comprises of a partition of the vertices and a directed graph defined over its components. CCPG satisfies consistency of orientations and additional constraints which favor finer partitions. Furthermore, it reduces to the underlying causal graph when the causal graph is identifiable. As a consequence, our results offer the first efficient algorithm for recovering the true causal graph with a polynomial number of tests, in special cases where the causal graph is fully identifiable through observational data and potentially additional interventions.

Foundation models have brought changes to the landscape of machine learning, demonstrating sparks of human-level intelligence across a diverse array of tasks. However, a gap persists in complex tasks such as causal inference, primarily due to challenges associated with intricate reasoning steps and high numerical precision requirements. In this work, we take a first step towards building causally-aware foundation models for treatment effect estimations. We propose a novel, theoretically justified method called Causal Inference with Attention (CInA), which utilizes multiple unlabeled datasets to perform self-supervised causal learning, and subsequently enables zero-shot causal inference on unseen tasks with new data. This is based on our theoretical results that demonstrate the primal-dual connection between optimal covariate balancing and self-attention, facilitating zero-shot causal inference through the final layer of a trained transformer-type architecture. We demonstrate empirically that CInA effectively generalizes to out-of-distribution datasets and various real-world datasets, matching or even surpassing traditional per-dataset methodologies. These results provide compelling evidence that our method has the potential to serve as a stepping stone for the development of causal foundation models.

Causal disentanglement aims to learn about latent causal factors behind data, hold- ing the promise to augment existing representation learning methods in terms of interpretability and extrapolation. Recent advances establish identifiability results assuming that interventions on (single) latent factors are available; however, it re- mains debatable whether such assumptions are reasonable due to the inherent nature of intervening on latent variables. Accordingly, we reconsider the fundamentals and ask what can be learned using just observational data. We provide a precise characterization of latent factors that can be identified in nonlinear causal models with additive Gaussian noise and linear mixing, without any interventions or graphical restrictions. In particular, we show that the causal variables can be identified up to a _layer_-wise transformation and that further disen- tanglement is not possible. We transform these theoretical results into a practical algorithm consisting of solving a quadratic program over the score estimation of the observed data. We provide simulation results to support our theoretical guarantees and demonstrate that our algorithm can derive meaningful causal representations from purely observational data.

Learning causal structures from interventional data is a fundamental problem with broad applications across various fields. While many previous works have focused on recovering the entire causal graph, in practice, there are scenarios where learning only part of the causal graph suffices. This is called \emph{targeted} causal discovery. In our work, we focus on two such well-motivated problems: subset search and causal matching. We aim to minimize the number of interventions in both cases. Towards this, we introduce the \emph{Meek separator}, which is a subset of vertices that, when intervened, decomposes the remaining unoriented edges into smaller connected components. We then present an efficient algorithm to find Meek separators that are of small sizes. Such a procedure is helpful in designing various divide-and-conquer-based approaches. In particular, we propose two randomized algorithms that achieve logarithmic approximation for subset search and causal matching, respectively. Our results provide the first known average-case provable guarantees for both problems. We believe that this opens up possibilities to design near-optimal methods for many other targeted causal structure learning problems arising from various applications.

Causal disentanglement aims to uncover a representation of data using latent variables that are interrelated through a causal model. Such a representation is identifiable if the latent model that explains the data is unique. In this paper, we focus on the scenario where unpaired observational and interventional data are available, with each intervention changing the mechanism of a latent variable. When the causal variables are fully observed, statistically consistent algorithms have been developed to identify the causal model under faithfulness assumptions. We here show that identifiability can still be achieved with unobserved causal variables, given a generalized notion of faithfulness. Our results guarantee that we can recover the latent causal model up to an equivalence class and predict the effect of unseen combinations of interventions, in the limit of infinite data. We implement our causal disentanglement framework by developing an autoencoding variational Bayes algorithm and apply it to the problem of predicting combinatorial perturbation effects in genomics.

In this paper, we consider the large-scale precoding matrix indicator (PMI) selection problem at the receiver in wireless communications. The selection is based on the channel capacity of the PMI matrix in a pre-designed codebook. The quality of the PMI matrix is essential in achieving higher spectral efficiency. We first derive two novel formulations including a partial permutation-matrix model and an indicator-vector model for the original problem. The discrete constraints in the formulations make the problem NP-hard. Then we propose a stochastic projected gradient method augmented by block coordinate descent under various strategies. We show that the algorithms terminate in finite steps and produce sufficient descent at each iteration when the step size is chosen properly. Extensive experiments demonstrate that our proposed algorithms are able to find better PMI matrices more efficiently compared to the existing methods.

Transforming a causal system from a given initial state to a desired target state is an important task permeating multiple fields including control theory, biology, and materials science. In causal models, such transformations can be achieved by performing a set of interventions. In this paper, we consider the problem of identifying a shift intervention that matches the desired mean of a system through active learning. We define the Markov equivalence class that is identifiable from shift interventions and propose two active learning strategies that are guaranteed to exactly match a desired mean. We then derive a worst-case lower bound for the number of interventions required and show that these strategies are optimal for certain classes of graphs. In particular, we show that our strategies may require exponentially fewer interventions than the previously considered approaches, which optimize for structure learning in the underlying causal graph. In line with our theoretical results, we also demonstrate experimentally that our proposed active learning strategies require fewer interventions compared to several baselines.