Probabilistic Integral Circuits
This work addresses the limitation of probabilistic circuits to discrete latent variables, enabling more expressive generative modeling while maintaining tractability, which is incremental but useful for machine learning practitioners in distribution estimation.
The paper introduces probabilistic integral circuits (PICs) to bridge the gap between tractable discrete probabilistic circuits and expressive continuous latent variable models, showing that PIC-approximating PCs outperform standard PCs on distribution estimation benchmarks.
Continuous latent variables (LVs) are a key ingredient of many generative models, as they allow modelling expressive mixtures with an uncountable number of components. In contrast, probabilistic circuits (PCs) are hierarchical discrete mixtures represented as computational graphs composed of input, sum and product units. Unlike continuous LV models, PCs provide tractable inference but are limited to discrete LVs with categorical (i.e. unordered) states. We bridge these model classes by introducing probabilistic integral circuits (PICs), a new language of computational graphs that extends PCs with integral units representing continuous LVs. In the first place, PICs are symbolic computational graphs and are fully tractable in simple cases where analytical integration is possible. In practice, we parameterise PICs with light-weight neural nets delivering an intractable hierarchical continuous mixture that can be approximated arbitrarily well with large PCs using numerical quadrature. On several distribution estimation benchmarks, we show that such PIC-approximating PCs systematically outperform PCs commonly learned via expectation-maximization or SGD.