Einsum Networks: Fast and Scalable Learning of Tractable Probabilistic Circuits

TL;DR

Einsum Networks (EiNets) offer a fast, scalable, and memory-efficient approach for probabilistic circuits, enabling effective training on complex datasets and serving as high-quality generative models.

Contribution

The paper introduces EiNets, a novel implementation of probabilistic circuits that significantly improves speed, memory efficiency, and scalability, and simplifies EM training via automatic differentiation.

Findings

EiNets achieve up to 100x speedup and memory savings over previous methods.

They successfully scale to large datasets like SVHN and CelebA.

EiNets function effectively as generative image models.

Abstract

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.