Distribution learning via neural differential equations: minimal energy regularization and approximation theory
This work provides theoretical guarantees for neural ODEs in generative modeling and density estimation, addressing a foundational issue in machine learning with potential broad impact.
The paper tackles the problem of approximating complex probability distributions using neural ODEs by showing that straight-line interpolations of transport maps can be realized with velocity fields minimized via a specific regularization, and it proves explicit bounds on network size needed to achieve desired accuracy in distribution approximation.
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 $ε> 0$ can be achieved by a deep neural network representation of the velocity field whose size is bounded explicitly in terms of $ε$, 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.