4.2 Regularizing Deterministic PCs

We demonstrate how the intuitive ideas for regularizing distributions presented in Sec. 2 (i.e., data
softening and entropy regularization) can be efﬁciently applied to deterministic PCs.
Data softening As hinted by Eq. (1), we need exponentially many samples to represent a softened
dataset, which makes parameter learning intractable even for the simple logistic regression model
(Thm. 1), let alone more complex probabilistic models such as VAEs [44] and GANs [45]. Despite

6

X1¬X1X2¬X2Z1¬Z1⇥⇥n1n2n3X1¬X1X2¬X2⇥⇥n1n2n3Z1Z2Z3X1Z4Z5X2X3X4X5HCLTX1X2X3X4X5CLTAlgorithm 3 PC Entropy regularization
1: Input: A deterministic PC p; ﬂow Fn,c(D) for every edge (n, c) in p; hyperparameter τ.
2: Output: A set of log-parameters, {ϕn,c : (n, c) ∈ p}, which are the solution of Eq. (2).
3: ∀n, node_prob[n] ← 0;
4: while not converge do
5:
6:
7:
8:
Tractable Regularization of Probabilistic Circuits