The aim in many sciences is to understand the mechanisms that underlie the observed distribution of variables, starting from a set of initial hypotheses. Causal discovery allows us to infer mechanisms as sets of cause and effect relationships in a generalized way -- without necessarily tailoring to a specific domain. Causal discovery algorithms search over a structured hypothesis space, defined by the set of directed acyclic graphs, to find the graph that best explains the data. For high-dimensional problems, however, this search becomes intractable and scalable algorithms for causal discovery are needed to bridge the gap. In this paper, we define a novel causal graph partition that allows for divide-and-conquer causal discovery with theoretical guarantees. We leverage the idea of a superstructure -- a set of learned or existing candidate hypotheses -- to partition the search space. We prove under certain assumptions that learning with a causal graph partition always yields the Markov Equivalence Class of the true causal graph. We show our algorithm achieves comparable accuracy and a faster time to solution for biologically-tuned synthetic networks and networks up to ${10^4}$ variables. This makes our method applicable to gene regulatory network inference and other domains with high-dimensional structured hypothesis spaces.

Given a classical channel, a stochastic map from inputs to outputs, can we replace the input with a simple intermediate variable that still yields the correct conditional output distribution? We examine two cases: first, when the intermediate variable is classical; second, when the intermediate variable is quantum. We show that the quantum variable's size is generically smaller than the classical, according to two different measures---cardinality and entropy. We demonstrate optimality conditions for a special case. We end with several related results: a proposal for extending the special case, a demonstration of the impact of quantum phases, and a case study concerning pure versus mixed states.

We provide a rigorous definition of the visual cause of a behavior that is broadly applicable to the visually driven behavior in humans, animals, neurons, robots and other perceiving systems. Our framework generalizes standard accounts of causal learning to settings in which the causal variables need to be constructed from micro-variables. We prove the Causal Coarsening Theorem, which allows us to gain causal knowledge from observational data with minimal experimental effort. The theorem provides a connection to standard inference techniques in machine learning that identify features of an image that correlate with, but may not cause, the target behavior. Finally, we propose an active learning scheme to learn a manipulator function that performs optimal manipulations on the image to automatically identify the visual cause of a target behavior. We illustrate our inference and learning algorithms in experiments based on both synthetic and real data.