Drifting Fields are not Conservative

2026-04-07

AI Summary

This work investigates whether the drift field in single-step generative models can be interpreted as the gradient of a scalar loss function—that is, as a conservative vector field. Theoretical analysis reveals that standard position-dependent normalizations, such as those based on Gaussian kernels, generally yield non-conservative drift fields, with the Gaussian kernel itself being a benign exception. To address this, the authors propose a sharp-kernel normalization technique that restores conservativeness for any radial kernel, thereby enabling the construction of an explicit training objective. Experimental results demonstrate that training with conservative drift fields performs on par with or better than non-conservative alternatives, offering a clearer and more principled optimization target for generative modeling.

Abstract

Drifting models generate high-quality samples in a single forward pass by transporting generated samples toward the data distribution using a vector valued drift field. We investigate whether this procedure is equivalent to optimizing a scalar loss and find that, in general, it is not: drift fields are not conservative - they cannot be written as the gradient of any scalar potential. We identify the position-dependent normalization as the source of non-conservatism. The Gaussian kernel is the unique exception where the normalization is harmless and the drift field is exactly the gradient of a scalar function. Generalizing this, we propose an alternative normalization via a related kernel (the sharp kernel) which restores conservatism for any radial kernel, yielding well-defined loss functions for training drifting models. While we identify that the drifting field matching objective is strictly more general than loss minimization, as it can implement non-conservative transport fields that no scalar loss can reproduce, we observe that practical gains obtained utilizing this flexibility are minimal. We thus propose to train drifting models with the conceptually simpler formulations utilizing loss functions.

Problem

• drifting fields
• conservative vector fields
• scalar potential
• normalization
• loss function

Innovation

• drifting fields
• non-conservative vector fields
• scalar potential
• kernel normalization
• sharp kernel