# Drifting Fields are not Conservative

arXiv: https://arxiv.org/abs/2604.06333 · HTML: https://mlsanity.com/paper/2604.06333/

## Key facts

- Cited by 13 papers in the map (data through 14 Oct 2026).
- Builds on 14 of its 25 references that are in the map.
- First posted on arXiv 7 Apr 2026.

## Builds on (most cited first, 14 of 14)

- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [High-Resolution Image Synthesis with Latent Diffusion Models](https://mlsanity.com/paper/2112.10752/) (2021)
- [Diffusion Models Beat GANs on Image Synthesis](https://mlsanity.com/paper/2105.05233/) (2021)
- [Scalable Diffusion Models with Transformers](https://mlsanity.com/paper/2212.09748/) (2022)
- [Classifier-Free Diffusion Guidance](https://mlsanity.com/paper/2207.12598/) (2022)
- [Fashion-MNIST: a Novel Image Dataset for Benchmarking Machine Learning Algorithms](https://mlsanity.com/paper/1708.07747/) (2017)
- [Improved Precision and Recall Metric for Assessing Generative Models](https://mlsanity.com/paper/1904.06991/) (2019)
- [Maximum Mean Discrepancy Gradient Flow](https://mlsanity.com/paper/1906.04370/) (2019)
- [Generative Modeling via Drifting](https://mlsanity.com/paper/2602.04770/) (2026)
- [Kernel Distribution Embeddings: Universal Kernels, Characteristic Kernels and Kernel Metrics on Distributions](https://mlsanity.com/paper/1604.05251/) (2016)
- [Sinkhorn-Drifting Generative Models](https://mlsanity.com/paper/2603.12366/) (2026)
- [Generative Drifting is Secretly Score Matching: a Spectral and Variational Perspective](https://mlsanity.com/paper/2603.09936/) (2026)
- [Gradient Flow Drifting: Generative Modeling via Wasserstein Gradient Flows of KDE-Approximated Divergences](https://mlsanity.com/paper/2603.10592/) (2026)
- [A Unified View of Score-Based and Drifting Models](https://mlsanity.com/paper/2603.07514/) (2026)

## Built on by (most cited first, 13 of 13)

- [On the Wasserstein Gradient Flow Interpretation of Drifting Models](https://mlsanity.com/paper/2605.05118/) (2026)
- [Kernel-Gradient Drifting Models](https://mlsanity.com/paper/2605.10727/) (2026)
- [Finite-Particle Convergence Rates for Conservative and Non-Conservative Drifting Models](https://mlsanity.com/paper/2605.22795/) (2026)
- [Three-Body Scattering for Generative Modeling](https://mlsanity.com/paper/2607.18198/) (2026)
- [DriftXpress: Faster Drifting Models via Projected RKHS Fields](https://mlsanity.com/paper/2605.12183/) (2026)
- [Uniform-in-time Propagation-of-Chaos for Stein Variational Gradient Descent](https://mlsanity.com/paper/2607.00149/) (2026)
- [Analytical Correction for Subsampling Bias in Drifting Models](https://mlsanity.com/paper/2604.27239/) (2026)
- [Unifying Distributional Training for One-Step Visual Generation](https://mlsanity.com/paper/2609.35763/) (2026)
- [Identifiability and Stability of Generative Drifting in the Companion-Elliptic Kernel Family](https://mlsanity.com/paper/2604.24196/) (2026)
- [Finite-Probe Total-Variation Certificates for Finite-Basis Drifting Models](https://mlsanity.com/paper/2608.01547/) (2026)
- [One-Step Generative Modeling via Unbalanced Optimal Transport](https://mlsanity.com/paper/2609.32708/) (2026)
- [One-Step Generative Modeling via Training Dynamics Action](https://mlsanity.com/paper/2610.00518/) (2026)
- [One-Step Generation via Riemannian Wasserstein Gradient Flows](https://mlsanity.com/paper/2610.04454/) (2026)

## Abstract (arXiv)

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.