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.

Ordinary differential equations (ODEs), via their induced flow maps, provide a powerful framework to parameterize invertible transformations for representing complex probability distributions. While such models have achieved enormous success in machine learning, little is known about their statistical properties. This work establishes the first general nonparametric statistical convergence analysis for distribution learning via ODE models trained through likelihood maximization. We first prove a convergence theorem applicable to arbitrary velocity field classes $\mathcal{F}$ satisfying certain simple boundary constraints. This general result captures the trade-off between the approximation error and complexity of the ODE model. We show that the latter can be quantified via the $C^1$-metric entropy of the class $\mathcal{F}$. We then apply this general framework to the setting of $C^k$-smooth target densities, and establish nearly minimax-optimal convergence rates for two relevant velocity field classes $\mathcal{F}$: $C^k$ functions and neural networks. The latter is the practically important case of neural ODEs. Our results also provide insight on how the choice of velocity field class, and the dependence of this choice on sample size (e.g., the scaling of neural network classes), impact statistical performance.

Graph-based semi-supervised learning is the problem of propagating labels from a small number of labelled data points to a larger set of unlabelled data. This paper is concerned with the consistency of optimization-based techniques for such problems, in the limit where the labels have small noise and the underlying unlabelled data is well clustered. We study graph-based probit for binary classification, and a natural generalization of this method to multi-class classification using one-hot encoding. The resulting objective function to be optimized comprises the sum of a quadratic form defined through a rational function of the graph Laplacian, involving only the unlabelled data, and a fidelity term involving only the labelled data. The consistency analysis sheds light on the choice of the rational function defining the optimization.