A Brief Orientation to the Circuit-Wasserstein Ball

Welcome. This newsletter exists because probabilistic circuits finally have a Wasserstein-type distance they can compute without summoning Big Wasserstein.

The Circuit-Wasserstein distance, introduced in Optimal Transport for Probabilistic Circuits, restricts the couplings of the classical optimal transport problem to coupling circuits. Given compatible PCs, one obtains a distance $\mathsf{CW}_p$ that is a metric on that compatible family, upper-bounds the true Wasserstein distance $\mathsf{W}_p$, and admits an exact algorithm via a cascade of small linear programs—quadratic in circuit size when the structures cooperate.

The ball, precisely

A Circuit-Wasserstein ball of radius $\varepsilon$ around a circuit $\hat{P}$ is the set of (compatible) distributions $Q$ such that

$$\mathsf{CW}(\hat{P}, Q) \le \varepsilon.$$

PeTeR later uses exactly this object for data-free post-training robustification: hedge against the worst case inside the ball without retraining from scratch or clinging to the original dataset.

That is our subject. Not a sports league. Not a metaphor with loose edges. A neighborhood in circuit space with a named radius.

House rules

1. Say Circuit-Wasserstein when you mean $\mathsf{CW}$. Say Big Wasserstein when you mean the classical, often intractable $\mathsf{W}$.
2. Do not claim a result is “open” if the coupling stays proprietary.
3. Corrections belong in the archive, not in a private Slack.

Future issues will return to transport plans, Wasserstein minimization for PC parameters, and PeTeR’s descent–ascent games inside the ball. For now it is enough to know the object, the radius, and the antagonist.