Elliptic Sampling: Optimal interpolation Schedules for Gaussians and Beyond

Generative modeling based on stochastic interpolants requires specifying two key choices: an interpolant schedule bridging the initial and target distributions, and an efficient discretization of the resulting probability-flow ODE. In this work, we introduce a new framework for joint schedule/discretization design based on the analyticity properties of the (complexified) ODE trajectories, which in turn ensure good numerical behavior. Our formulation can be solved exactly via conformal mapping, yielding a natural family of variance-preserving elliptic schedules, parametrized by Jacobi elliptic functions. Pairing an elliptic schedule with a tailored discretization, we obtain Zoloflow, a new sampling algorithm closely connected to Zolotarev’s optimal rational function approximants. Crucially, both the elliptic schedule and Zoloflow explicitly incorporate the condition number $\kappa$ of the target distribution and reduce to the classical trig schedule in the limit $\kappa \to 1$. For Gaussian targets, Zoloflow achieves minimax optimality among all oblivious score-span algorithms, attaining total-variation error $\epsilon$ using $\Theta (\log \kappa \log (\sqrt{d}/\epsilon ))$ (exact) score queries. For the log-concave case, the elliptic schedule allows sampling with at most $\epsilon$ 2-Wasserstein-error by requiring merely $\tilde{O}(\sqrt{d}\mathrm{polylog}(\kappa ,1/\epsilon ))$ score evaluations. Complementing these theoretical results, we empirically evaluate the elliptic schedule on general (non-log-concave) target distributions, including detailed comparisons with alternative schedule proposals on both synthetic benchmarks and existing image generation datasets.