Distribution learning via neural differential equations: minimal energy regularization and approximation theory

TL;DR

This paper demonstrates that neural ODEs can effectively approximate complex probability distributions through minimal energy regularization, providing explicit bounds and guarantees for distribution learning in generative modeling.

Contribution

It introduces a novel regularization approach for neural ODEs that minimizes energy, along with explicit bounds and stability guarantees for distribution approximation.

Findings

Existence of velocity fields realizing straight-line interpolation for transport maps.

Explicit polynomial bounds for the $C^k$ norm of velocity fields.

Achievability of distribution approximation with neural networks of bounded size.

Abstract

Neural ordinary differential equations (ODEs) provide expressive representations of invertible transport maps that can be used to approximate complex probability distributions, e.g., for generative modeling, density estimation, and Bayesian inference. We show that for a large class of transport maps $T$, there exists a time-dependent ODE velocity field realizing a straight-line interpolation $(1-t)x + tT(x)$, $t \in [0,1]$, of the displacement induced by the map. Moreover, we show that such velocity fields are minimizers of a training objective containing a specific minimum-energy regularization. We then derive explicit upper bounds for the $C^k$ norm of the velocity field that are polynomial in the $C^k$ norm of the corresponding transport map $T$; in the case of triangular (Knothe--Rosenblatt) maps, we also show that these bounds are polynomial in the $C^k$ norms of the associated source and target densities. Combining these results with stability arguments for distribution approximation via ODEs, we show that Wasserstein or Kullback--Leibler approximation of the target distribution to any desired accuracy $\epsilon > 0$ can be achieved by a deep neural network representation of the velocity field whose size is bounded explicitly in terms of $\epsilon$, the dimension, and the smoothness of the source and target densities. The same neural network ansatz yields guarantees on the value of the regularized training objective.