  We derive a set of causal deep neural networks whose architectures are a
consequence of tensor (multilinear) factor analysis. Forward causal questions
are addressed with a neural network architecture composed of causal capsules
and a tensor transformer. The former estimate a set of latent variables that
represent the causal factors, and the latter governs their interaction. Causal
capsules and tensor transformers may be implemented using shallow autoencoders,
but for a scalable architecture we employ block algebra and derive a deep
neural network composed of a hierarchy of autoencoders. An interleaved kernel
hierarchy preprocesses the data resulting in a hierarchy of kernel tensor
factor models. Inverse causal questions are addressed with a neural network
that implements multilinear projection 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 are well-defined and
produce multiple candidate solutions. Our forward and inverse neural network
architectures are suitable for asynchronous parallel computation.
