# Coarse-Grained Kullback--Leibler Control of Diffusion-Based Generative AI

arXiv: https://arxiv.org/abs/2601.01045 · HTML: https://mlsanity.com/paper/2601.01045/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Placed in Diffusion foundations and sampling, one of 14 directions in Diffusion models.
- Builds on 10 of its 26 references that are in the map.
- First posted on arXiv 3 Jan 2026.

## Placed in

- Diffusion models / [Diffusion foundations and sampling](https://mlsanity.com/diffusion/diffusion-foundations-and-sampling/)

## Builds on (most cited first, 10 of 10)

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Improved Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2102.09672/) (2021)
- [Improved Techniques for Training GANs](https://mlsanity.com/paper/1606.03498/) (2016)
- [Diffusion Schr"odinger Bridge with Applications to Score-Based Generative Modeling](https://mlsanity.com/paper/2106.01357/) (2021)
- [Deep Generative Learning via Schr"odinger Bridge](https://mlsanity.com/paper/2106.10410/) (2021)
- [Conditional sampling within generative diffusion models](https://mlsanity.com/paper/2409.09650/) (2024)
- [Safe and Stable Control via Lyapunov-Guided Diffusion Models](https://mlsanity.com/paper/2509.25375/) (2025)
- [Generative diffusion models from a PDE perspective](https://mlsanity.com/paper/2501.17054/) (2025)
- [A PDE Perspective on Generative Diffusion Models](https://mlsanity.com/paper/2511.05940/) (2025)

## Abstract (arXiv)

Diffusion models and score-based generative models provide a powerful framework for synthesizing high-quality images from noise. However, there is still no satisfactory theory that describes how coarse-grained quantities, such as blockwise intensity or class proportions after partitioning an image into spatial blocks, are preserved and evolve along the reverse diffusion dynamics. In previous work, the author introduced an information-theoretic Lyapunov function V for non-ergodic Markov processes on a state space partitioned into blocks, defined as the minimal Kullback-Leibler divergence to the set of stationary distributions reachable from a given initial condition, and showed that a leak-tolerant potential V-delta with a prescribed tolerance for block masses admits a closed-form expression as a scaling-and-clipping operation on block masses. In this paper, I transplant this framework to the reverse diffusion process in generative models and propose a reverse diffusion scheme that is projected by the potential V-delta (referred to as the V-delta projected reverse diffusion). I extend the monotonicity of V to time-inhomogeneous block-preserving Markov kernels and show that, under small leakage and the V-delta projection, V-delta acts as an approximate Lyapunov function. Furthermore, using a toy model consisting of block-constant images and a simplified reverse kernel, I numerically demonstrate that the proposed method keeps the block-mass error and the leak-tolerant potential within the prescribed tolerance, while achieving pixel-wise accuracy and visual quality comparable to the non-projected dynamics. This study reinterprets generative sampling as a decrease of an information potential from noise to data, and provides a design principle for reverse diffusion processes with explicit control of coarse-grained quantities.