The kernels automatically segment PC units into layers such that the computation in every layer can
be fully parallelized. As a result, we can train PCs with millions of parameters in less than an hour
and en- or decode samples very efﬁciently. Implementation details can be found in Appx. B.3.
Related work Finding good PC architectures has been a central topic in the literature (Choi et al.,
2020). A recent trend for learning smooth and (structured-)decomposable PCs is to construct large
models with pre-deﬁned architecture, which is mainly determined by the variable ordering strategies.
For example, RAT-SPNs (Peharz et al., 2020b) and EiNets (Peharz et al., 2020a) use random variable
orders, Gens & Domingos (2013) proposes an effective variable ordering for image data, and other
works propose data-dependent orderings based on certain information criterion (Rooshenas & Lowd,
2014) or clustering algorithms (Gens & Domingos, 2013). Alternatively, researchers have focused
LOSSLESS COMPRESSION WITH PROBABILISTIC CIRCUITS