The Zero-Flow Phenomena

Theorem 3.1 (zero-flow criterion). Let \(v_t\) be the rectified flow trained with independent coupling between \(p_{\mathrm{source}}\) and \(p_{\mathrm{target}}\). Then \(v_{1/2} \equiv 0\) everywhere if and only if \(p_{\mathrm{source}} = p_{\mathrm{target}}\).

A rectified flow is trained with independent coupling between the two distributions you choose; particles are then transported from \(t = 0\) to \(t = 1\) along the learned velocity field. Pick the same dataset twice and watch everything freeze blue at the midpoint.

Theorem 3.4 (conditional zero-flow condition). For all pairs \((\xi, \eta)\) such that \(f(\xi) = \eta\), and for all \(z\), the velocity field \(v_{1/2}(z;\, \eta,\, \xi) = 0\) if and only if \(p_{X|Y} = p_{X|f(Y)}\).

Data follow the joint model \(Y \sim \mathcal{N}(0,1)\), \(X = 0.5\,Y + \varepsilon\), \(\varepsilon \sim \mathcal{N}(0,1)\). A conditional rectified flow \(u_t(x;\, f(Y'),\, Y)\) is trained with independent coupling between \((X, Y)\) and an independent copy \((X', Y')\); each data point \((Y_i, X_i)\) is then transported vertically along the learned field. The midpoint field vanishes if and only if the statistic \(f\) is sufficient for predicting \(X\). Choose \(f\) below: a sufficient statistic, e.g. \(\sigma(-2y)\), freezes everything blue at \(t = \tfrac{1}{2}\); a lossy one, e.g. \(\sin(2y)\), does not.