COAL: Jacobian-Free Likelihood Evaluation for Rectified Flows

Flow-based generative models achieve strong sample quality, but per-sample likelihood evaluation under the instantaneous change-of-variables formula remains bottlenecked by an $\mathcal{O}(d^3)$ divergence integral. Hutchinson's stochastic trace estimator partially mitigates this, at the cost of backpropagations through the velocity network per ODE step and a noisy integrand. We show that the divergence of a Rectified Flow velocity admits a deterministic closed form: by Tweedie's formula, it equals the trace of the conditional covariance of the velocity residual, scaled by a known function of. We amortize this trace with a lightweight per-coordinate variance head trained jointly with the velocity, yielding Coal (covariance-amortized likelihood): a deterministic per-sample likelihood estimator that runs in a single forward pass per ODE step, with no input-space backpropagation and no stochastic probes. On class-conditional ImageNet with a DiT-XL latent rectified flow, Coal matches the Hutchinson reference at less evaluation compute.