  Probabilistic integral circuits (PICs) have been recently introduced as
probabilistic models enjoying the key ingredient behind expressive generative
models: continuous latent variables (LVs). PICs are symbolic computational
graphs defining continuous LV models as hierarchies of functions that are
summed and multiplied together, or integrated over some LVs. They are tractable
if LVs can be analytically integrated out, otherwise they can be approximated
by tractable probabilistic circuits (PC) encoding a hierarchical numerical
quadrature process, called QPCs.
  So far, only tree-shaped PICs have been explored, and training them via
numerical quadrature requires memory-intensive processing at scale. In this
paper, we address these issues, and present: (i) a pipeline for building
DAG-shaped PICs out of arbitrary variable decompositions, (ii) a procedure for
training PICs using tensorized circuit architectures, and (iii) neural
functional sharing techniques to allow scalable training. In extensive
experiments, we showcase the effectiveness of functional sharing and the
superiority of QPCs over traditional PCs.
