Drifting Fields are not Conservative

TL;DR

This paper investigates the nature of drifting models' vector fields, revealing they are generally non-conservative, and introduces a sharp normalization technique to make them conservative, enabling direct optimization.

Contribution

The paper identifies the non-conservatism in drifting models' vector fields and proposes a sharp normalization method to produce conservative fields for improved optimization.

Findings

Sharp normalization preserves original performance.

Gaussian kernel is the unique radial exception.

The resulting vector field is the gradient of a scalar potential.

Abstract

Drifting models have recently gained attention for generating high-quality samples in a single forward pass. During training, they learn a push-forward map by following a vector-valued field, the drift field. We ask whether this procedure is equivalent to optimizing a scalar loss and find that, in general, it is not: drift fields are not conservative and cannot be written as the gradient of any scalar potential. We identify the position-dependent normalization as the source of non-conservatism, with the Gaussian kernel as the unique radial exception. Guided by this, we introduce the sharp kernel $k^\#$ and a sharp-normalized drift field that is conservative for general radial kernels. The resulting vector field is the gradient of a scalar potential that can be optimized directly using stochastic gradient descent. Moreover, the field has the form of a score difference of kernel density estimates, and gives exact equilibrium identifiability. Thus, sharp normalization closes the gap to related literature, such as Wasserstein gradient-flows and denoising score matching, also for non-Gaussian kernels. Empirically, sharp normalization preserves the performance of the original drifting objective, suggesting that the non-conservative flexibility is not required for high-quality generation.