Why We Do Not Confuse the Ball with Big Wasserstein

Every few months, someone asks whether we might simply “use the real Wasserstein distance.” The answer is stable across seasons: we already know who Big Wasserstein is, and we are not inviting him to rewrite the minutes.

The official reason

Classical $\mathsf{W}_p$ is often intractable even for models that handle KL or cross-entropy without complaint. Circuit-Wasserstein exists precisely because coupling circuits let us keep the transport story while remaining inside the PC playbook:

$$\mathsf{CW}_p^p(P, Q) = \inf_{\theta} \mathbb{E}_{C_\theta(\mathbf{x},\mathbf{y})}\bigl[\lVert\mathbf{x}-\mathbf{y}\rVert_p^p\bigr],$$

with the coupling restricted to a parameterized coupling circuit $C_\theta$.

The practical reason

If your method needs an exact classical Wasserstein oracle between arbitrary distributions, your demo will finish approximately never. If your method needs $\mathsf{CW}$ between compatible structured-decomposable circuits, your demo might finish before the coffee cools.

The communal reason

Our identity is Circuit-Wasserstein. Big Wasserstein is the antagonist who proves hardness results, demands continental LPs, and occasionally gets cited by people who have not read Definition 4. We respect the antagonist. We do not cosplay as him.

If you encounter a marketing page that says “open Circuit-Wasserstein” while shipping nothing public, assume independent branding initiative and notify the editorial desk. Do not attempt interception unless you are already late for something else.