  Probabilistic circuits are a unifying representation of functions as
computation graphs of weighted sums and products. Their primary application is
in probabilistic modeling, where circuits with non-negative weights (monotone
circuits) can be used to represent and learn density/mass functions, with
tractable marginal inference. Recently, it was proposed to instead represent
densities as the square of the circuit function (squared circuits); this allows
the use of negative weights while retaining tractability, and can be
exponentially more compact than monotone circuits. Unfortunately, we show the
reverse also holds, meaning that monotone circuits and squared circuits are
incomparable in general. This raises the question of whether we can reconcile,
and indeed improve upon the two modeling approaches. We answer in the positive
by proposing InceptionPCs, a novel type of circuit that naturally encompasses
both monotone circuits and squared circuits as special cases, and employs
complex parameters. Empirically, we validate that InceptionPCs can outperform
both monotone and squared circuits on image datasets.
