  Probabilistic circuits (PCs) are a promising avenue for probabilistic
modeling, as they permit a wide range of exact and efficient inference
routines. Recent ``deep-learning-style'' implementations of PCs strive for a
better scalability, but are still difficult to train on real-world data, due to
their sparsely connected computational graphs. In this paper, we propose Einsum
Networks (EiNets), a novel implementation design for PCs, improving prior art
in several regards. At their core, EiNets combine a large number of arithmetic
operations in a single monolithic einsum-operation, leading to speedups and
memory savings of up to two orders of magnitude, in comparison to previous
implementations. As an algorithmic contribution, we show that the
implementation of Expectation-Maximization (EM) can be simplified for PCs, by
leveraging automatic differentiation. Furthermore, we demonstrate that EiNets
scale well to datasets which were previously out of reach, such as SVHN and
CelebA, and that they can be used as faithful generative image models.
