Optimal Scheduling of Dynamic Transport
arXiv:2504.14425v1 Announce Type: new
Abstract: Flow-based methods for sampling and generative modeling use continuous-time dynamical systems to represent a {transport map} that pushes forward a source measure to a target measure. The introduction of a time axis provides considerable design freedom, and a central question is how to exploit this freedom. Though many popular methods seek straight line (i.e., zero acceleration) trajectories, we show here that a specific class of “curved” trajectories can significantly improve approximation and learning. In particular, we consider the unit-time interpolation of any given transport map $T$ and seek the schedule $tau: [0,1] to [0,1]$ that minimizes the spatial Lipschitz constant of the corresponding velocity field over all times $t in [0,1]$. This quantity is crucial as it allows for control of the approximation error when the velocity field is learned from data. We show that, for a broad class of source/target measures and transport maps $T$, the emph{optimal schedule} can be computed in closed form, and that the resulting optimal Lipschitz constant is emph{exponentially smaller} than that induced by an identity schedule (corresponding to, for instance, the Wasserstein geodesic). Our proof technique relies on the calculus of variations and $Gamma$-convergence, allowing us to approximate the aforementioned degenerate objective by a family of smooth, tractable problems.
Abstract: Flow-based methods for sampling and generative modeling use continuous-time dynamical systems to represent a {transport map} that pushes forward a source measure to a target measure. The introduction of a time axis provides considerable design freedom, and a central question is how to exploit this freedom. Though many popular methods seek straight line (i.e., zero acceleration) trajectories, we show here that a specific class of “curved” trajectories can significantly improve approximation and learning. In particular, we consider the unit-time interpolation of any given transport map $T$ and seek the schedule $tau: [0,1] to [0,1]$ that minimizes the spatial Lipschitz constant of the corresponding velocity field over all times $t in [0,1]$. This quantity is crucial as it allows for control of the approximation error when the velocity field is learned from data. We show that, for a broad class of source/target measures and transport maps $T$, the emph{optimal schedule} can be computed in closed form, and that the resulting optimal Lipschitz constant is emph{exponentially smaller} than that induced by an identity schedule (corresponding to, for instance, the Wasserstein geodesic). Our proof technique relies on the calculus of variations and $Gamma$-convergence, allowing us to approximate the aforementioned degenerate objective by a family of smooth, tractable problems.
Panos Tsimpos, Zhi Ren, Jakob Zech, Youssef Marzouk
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