Causal Deep Learning

We derive a set of causal deep neural networks whose architectures are a consequence of tensor (multilinear) factor analysis, a framework that facilitates causal inference. Forward causal questions are addressed with a neural network architecture composed of causal capsules and a tensor transformer. Causal capsules compute a set of invariant causal factor representations, whose interactions are governed by a tensor transformation. Inverse causal questions are addressed with a neural network that implements the multilinear projection algorithm. The architecture reverses the order of operations of a forward neural network and estimates the causes of effects. As an alternative to aggressive bottleneck dimension reduction or regularized regression that may camouflage an inherently underdetermined inverse problem, we prescribe modeling different aspects of the mechanism of data formation with piecewise tensor models whose multilinear projections produce multiple candidate solutions. Our forward and inverse questions may be addressed with shallow architectures, but for computationally scalable solutions, we derive a set of deep neural networks by taking advantage of block algebra. An interleaved kernel hierarchy results in doubly non-linear tensor factor models. The causal neural networks that are a consequence of tensor factor analysis are data agnostic, but are illustrated with facial images. Sequential, parallel and asynchronous parallel computation strategies are described.