Conservation Laws Characterize Likelihood in Diffusion Models.

Ziv Aharoni, Henry D. Pfister· July 14, 2026 View original

Summary

This paper develops conservation laws for diffusion models based on generalized extrinsic information transfer (GEXIT) functions, showing that data-model cross-entropy can be precisely characterized as an integral of local information-theoretic derivatives. This unified framework explains likelihood for discrete and continuous diffusion, with implications for training by minimizing negative log-likelihood.

Researchers have established a set of conservation laws for diffusion models, drawing parallels with autoregressive models' exact likelihood optimization. These laws are based on generalized extrinsic information transfer (GEXIT) functions and demonstrate that the data-model cross-entropy can be precisely defined as an integral of local information-theoretic derivatives along the noise path. This framework offers a unified characterization of likelihood for both discrete and continuous diffusion processes, with the Gaussian case aligning with the well-known mutual information-minimum mean-square error (I-MMSE) relationship. A key implication is a locality property, meaning these information-theoretic derivatives can be computed using only marginal posteriors along the noise path. Consequently, training diffusion models can be simplified to learning these marginal posteriors by minimizing the negative log-likelihood. The study also notes that while entropy is path-independent, finite-capacity denoisers approximate posteriors differently across noise types, leading to performance variations. These theoretical predictions are validated on synthetic Markov sources and standard benchmarks.

Why it matters

AI researchers and engineers working with generative models can gain a deeper theoretical understanding of diffusion models, potentially leading to more principled training objectives, improved likelihood estimation, and better model performance.

How to implement this in your domain

  1. 1Study the theoretical framework of conservation laws and GEXIT functions for diffusion models.
  2. 2Explore how to reformulate diffusion model training objectives to explicitly minimize negative log-likelihood based on marginal posteriors.
  3. 3Investigate the impact of different noise processes and denoiser capacities on model performance in light of the locality property.
  4. 4Apply the insights to design more efficient or robust training strategies for custom diffusion models.
  5. 5Benchmark new training approaches against traditional denoising objectives on relevant generative tasks.

Who benefits

AI/ML DevelopmentResearch & DevelopmentContent Creation (AI-generated media)Scientific Computing

Key takeaways

  • Conservation laws provide a unified likelihood characterization for diffusion models.
  • Data-model cross-entropy can be expressed as an integral of local information-theoretic derivatives.
  • Training can be simplified to minimizing negative log-likelihood by learning marginal posteriors.
  • Entropy is noise-path independent, but denoiser capacity affects performance.

Original post by Ziv Aharoni, Henry D. Pfister

"arXiv:2607.10067v1 Announce Type: new Abstract: While autoregressive models optimize the exact data likelihood via the chain rule, diffusion models are typically trained with denoising objectives. We develop conservation laws based on generalized extrinsic information transfer (G…"

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