Deep generative models (DGMs) have recently demonstrated remarkable success in capturing complex probability distributions over graphs. Although their excellent performance is attributed to powerful and scalable deep neural networks, it is, at the same time, exactly the presence of these highly non-linear transformations that makes DGMs intractable. Indeed, despite representing probability distributions, intractable DGMs deny probabilistic foundations by their inability to answer even the most basic inference queries without approximations or design choices specific to a very narrow range of queries. To address this limitation, we propose probabilistic graph circuits (PGCs), a framework of tractable DGMs that provide exact and efficient probabilistic inference over (arbitrary parts of) graphs. Nonetheless, achieving both exactness and efficiency is challenging in the permutation-invariant setting of graphs. We design PGCs that are inherently invariant and satisfy these two requirements, yet at the cost of low expressive power. Therefore, we investigate two alternative strategies to achieve the invariance: the first sacrifices the efficiency, and the second sacrifices the exactness. We demonstrate that ignoring the permutation invariance can have severe consequences in anomaly detection, and that the latter approach is competitive with, and sometimes better than, existing intractable DGMs in the context of molecular graph generation.

Zhang et al. (ICML 2021, PLMR 139, pp. 12447-1245) introduced probabilistic generating circuits (PGCs) as a probabilistic model to unify probabilistic circuits (PCs) and determinantal point processes (DPPs). At a first glance, PGCs store a distribution in a very different way, they compute the probability generating polynomial instead of the probability mass function and it seems that this is the main reason why PGCs are more powerful than PCs or DPPs. However, PGCs also allow for negative weights, whereas classical PCs assume that all weights are nonnegative. One of the main insights of our paper is that the negative weights are responsible for the power of PGCs and not the different representation. PGCs are PCs in disguise, in particular, we show how to transform any PGC into a PC with negative weights with only polynomial blowup. PGCs were defined by Zhang et al. only for binary random variables. As our second main result, we show that there is a good reason for this: we prove that PGCs for categorial variables with larger image size do not support tractable marginalization unless NP = P. On the other hand, we show that we can model categorial variables with larger image size as PC with negative weights computing set-multilinear polynomials. These allow for tractable marginalization. In this sense, PCs with negative weights strictly subsume PGCs.

Generating functions, which are widely used in combinatorics and probability theory, encode function values into the coefficients of a polynomial. In this paper, we explore their use as a tractable probabilistic model, and propose probabilistic generating circuits (PGCs) for their efficient representation. PGCs strictly subsume many existing tractable probabilistic models, including determinantal point processes (DPPs), probabilistic circuits (PCs) such as sum-product networks, and tractable graphical models. We contend that PGCs are not just a theoretical framework that unifies vastly different existing models, but also show huge potential in modeling realistic data. We exhibit a simple class of PGCs that are not trivially subsumed by simple combinations of PCs and DPPs, and obtain competitive performance on a suite of density estimation benchmarks. We also highlight PGCs' connection to the theory of strongly Rayleigh distributions.

In recent years, printable graphical codes have attracted a lot of attention enabling a link between the physical and digital worlds, which is of great interest for the IoT and brand protection applications. The security of printable codes in terms of their reproducibility by unauthorized parties or clonability is largely unexplored. In this paper, we try to investigate the clonability of printable graphical codes from a machine learning perspective. The proposed framework is based on a simple system composed of fully connected neural network layers. The results obtained on real codes printed by several printers demonstrate a possibility to accurately estimate digital codes from their printed counterparts in certain cases. This provides a new insight on scenarios, where printable graphical codes can be accurately cloned.