# Efficiently Access Diffusion Fisher: Within the Outer Product Span Space

arXiv: https://arxiv.org/abs/2505.23264 · HTML: https://mlsanity.com/paper/2505.23264/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Placed in Diffusion foundations and sampling, one of 14 directions in Diffusion models.
- Builds on 48 of its 66 references that are in the map.
- First posted on arXiv 29 May 2025.

## Placed in

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

## Builds on (most cited first, 30 of 48)

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [High-Resolution Image Synthesis with Latent Diffusion Models](https://mlsanity.com/paper/2112.10752/) (2021)
- [Microsoft COCO: Common Objects in Context](https://mlsanity.com/paper/1405.0312/) (2014)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Diffusion Models Beat GANs on Image Synthesis](https://mlsanity.com/paper/2105.05233/) (2021)
- [Deep Unsupervised Learning using Nonequilibrium Thermodynamics](https://mlsanity.com/paper/1503.03585/) (2015)
- [Hierarchical Text-Conditional Image Generation with CLIP Latents](https://mlsanity.com/paper/2204.06125/) (2022)
- [Photorealistic Text-to-Image Diffusion Models with Deep Language Understanding](https://mlsanity.com/paper/2205.11487/) (2022)
- [Classifier-Free Diffusion Guidance](https://mlsanity.com/paper/2207.12598/) (2022)
- [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022)
- [LAION-5B: An open large-scale dataset for training next generation image-text models](https://mlsanity.com/paper/2210.08402/) (2022)
- [Generative Modeling by Estimating Gradients of the Data Distribution](https://mlsanity.com/paper/1907.05600/) (2019)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [GLIDE: Towards Photorealistic Image Generation and Editing with Text-Guided Diffusion Models](https://mlsanity.com/paper/2112.10741/) (2021)
- [Elucidating the Design Space of Diffusion-Based Generative Models](https://mlsanity.com/paper/2206.00364/) (2022)
- [DPM-Solver: A Fast ODE Solver for Diffusion Probabilistic Model Sampling in Around 10 Steps](https://mlsanity.com/paper/2206.00927/) (2022)
- [Cascaded Diffusion Models for High Fidelity Image Generation](https://mlsanity.com/paper/2106.15282/) (2021)
- [Variational Diffusion Models](https://mlsanity.com/paper/2107.00630/) (2021)
- [Stochastic Interpolants: A Unifying Framework for Flows and Diffusions](https://mlsanity.com/paper/2303.08797/) (2023)
- [DPM-Solver++: Fast Solver for Guided Sampling of Diffusion Probabilistic Models](https://mlsanity.com/paper/2211.01095/) (2022)
- [Pseudo Numerical Methods for Diffusion Models on Manifolds](https://mlsanity.com/paper/2202.09778/) (2022)
- [Maximum Likelihood Training of Score-Based Diffusion Models](https://mlsanity.com/paper/2101.09258/) (2021)
- [Analytic-DPM: an Analytic Estimate of the Optimal Reverse Variance in Diffusion Probabilistic Models](https://mlsanity.com/paper/2201.06503/) (2022)
- [A note on the evaluation of generative models](https://mlsanity.com/paper/1511.01844/) (2015)
- [EDICT: Exact Diffusion Inversion via Coupled Transformations](https://mlsanity.com/paper/2211.12446/) (2022)
- [Nearly d-Linear Convergence Bounds for Diffusion Models via Stochastic Localization](https://mlsanity.com/paper/2308.03686/) (2023)
- [Estimating the Optimal Covariance with Imperfect Mean in Diffusion Probabilistic Models](https://mlsanity.com/paper/2206.07309/) (2022)
- [Maximum Likelihood Training for Score-Based Diffusion ODEs by High-Order Denoising Score Matching](https://mlsanity.com/paper/2206.08265/) (2022)
- [Improved Techniques for Maximum Likelihood Estimation for Diffusion ODEs](https://mlsanity.com/paper/2305.03935/) (2023)
- [Theoretical Insights for Diffusion Guidance: A Case Study for Gaussian Mixture Models](https://mlsanity.com/paper/2403.01639/) (2024)

## Built on by (most cited first, 5 of 5)

- [FG-OrIU: Towards Better Forgetting via Feature-Gradient Orthogonality for Incremental Unlearning](https://mlsanity.com/paper/2601.13578/) (2026)
- [Unleashing High-Quality Image Generation in Diffusion Sampling Using Second-Order Levenberg-Marquardt-Langevin](https://mlsanity.com/paper/2505.24222/) (2025)
- [Likelihood Matching for Diffusion Models](https://mlsanity.com/paper/2508.03636/) (2025)
- [Analyzing and Improving Diffusion Models for Time-Series Data Imputation: A Proximal Recursion Perspective](https://mlsanity.com/paper/2602.01182/) (2026)
- [Fisher-Preserving Guidance: Training-Free Manifold Constraints for Safe Diffusion Control](https://mlsanity.com/paper/2605.29937/) (2026)

## Abstract (arXiv)

Recent Diffusion models (DMs) advancements have explored incorporating the second-order diffusion Fisher information (DF), defined as the negative Hessian of log density, into various downstream tasks and theoretical analysis. However, current practices typically approximate the diffusion Fisher by applying auto-differentiation to the learned score network. This black-box method, though straightforward, lacks any accuracy guarantee and is time-consuming. In this paper, we show that the diffusion Fisher actually resides within a space spanned by the outer products of score and initial data. Based on the outer-product structure, we develop two efficient approximation algorithms to access the trace and matrix-vector multiplication of DF, respectively. These algorithms bypass the auto-differentiation operations with time-efficient vector-product calculations. Furthermore, we establish the approximation error bounds for the proposed algorithms. Experiments in likelihood evaluation and adjoint optimization demonstrate the superior accuracy and reduced computational cost of our proposed algorithms. Additionally, based on the novel outer-product formulation of DF, we design the first numerical verification experiment for the optimal transport property of the general PF-ODE deduced map.