# On the spectral properties of generative denoiser Jacobians

arXiv: https://arxiv.org/abs/2609.36210 · HTML: https://mlsanity.com/paper/2609.36210/

## 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 29 of its 35 references that are in the map.
- First posted on arXiv 28 Sep 2026.

## Placed in

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

## Builds on (most cited first, 29 of 29)

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [High-Resolution Image Synthesis with Latent Diffusion Models](https://mlsanity.com/paper/2112.10752/) (2021)
- [Denoising Diffusion Implicit Models](https://mlsanity.com/paper/2010.02502/) (2020)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [ImageNet Large Scale Visual Recognition Challenge](https://mlsanity.com/paper/1409.0575/) (2014)
- [Deep Unsupervised Learning using Nonequilibrium Thermodynamics](https://mlsanity.com/paper/1503.03585/) (2015)
- [Scalable Diffusion Models with Transformers](https://mlsanity.com/paper/2212.09748/) (2022)
- [Classifier-Free Diffusion Guidance](https://mlsanity.com/paper/2207.12598/) (2022)
- [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022)
- [Scaling Rectified Flow Transformers for High-Resolution Image Synthesis](https://mlsanity.com/paper/2403.03206/) (2024)
- [Improved Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2102.09672/) (2021)
- [Elucidating the Design Space of Diffusion-Based Generative Models](https://mlsanity.com/paper/2206.00364/) (2022)
- [SiT: Exploring Flow and Diffusion-based Generative Models with Scalable Interpolant Transformers](https://mlsanity.com/paper/2401.08740/) (2024)
- [Representation Alignment for Generation: Training Diffusion Transformers Is Easier Than You Think](https://mlsanity.com/paper/2410.06940/) (2024)
- [Assessing Generative Models via Precision and Recall](https://mlsanity.com/paper/1806.00035/) (2018)
- [On the Importance of Noise Scheduling for Diffusion Models](https://mlsanity.com/paper/2301.10972/) (2023)
- [Generalization in diffusion models arises from geometry-adaptive harmonic representations](https://mlsanity.com/paper/2310.02557/) (2023)
- [Fast Timing-Conditioned Latent Audio Diffusion](https://mlsanity.com/paper/2402.04825/) (2024)
- [Generalized Denoising Auto-Encoders as Generative Models](https://mlsanity.com/paper/1305.6663/) (2013)
- [Input Perturbation Reduces Exposure Bias in Diffusion Models](https://mlsanity.com/paper/2301.11706/) (2023)
- [What matters for Representation Alignment: Global Information or Spatial Structure?](https://mlsanity.com/paper/2512.10794/) (2025)
- [Diffuse and Disperse: Image Generation with Representation Regularization](https://mlsanity.com/paper/2506.09027/) (2025)
- [On the Posterior Distribution in Denoising: Application to Uncertainty Quantification](https://mlsanity.com/paper/2309.13598/) (2023)
- [Self-Supervised Flow Matching for Scalable Multi-Modal Synthesis](https://mlsanity.com/paper/2603.06507/) (2026)
- [Contrastive Flow Matching](https://mlsanity.com/paper/2506.05350/) (2025)
- [Locality in Image Diffusion Models Emerges from Data Statistics](https://mlsanity.com/paper/2509.09672/) (2025)
- [Nuclear Norm Regularization for Deep Learning](https://mlsanity.com/paper/2405.14544/) (2024)
- [Fast constrained sampling in pre-trained diffusion models](https://mlsanity.com/paper/2410.18804/) (2024)

## Abstract (arXiv)

Generative denoising models, such as diffusion and flow-matching, learn to sample from complex distributions by training a deep neural network denoiser to recover clean data from noise-corrupted samples. While such models are typically compared on the quality of their synthesized samples, these metrics provide limited insight into how the underlying denoiser, which drives generation, differs. In this work, we propose to analyze the spectrum of the denoiser Jacobian as a tool to characterize these differences. Across pre-trained denoising models, we observe that better generative performance is associated with larger Jacobian eigenvalues. Motivated by this, we introduce a regularization scheme that controls the Jacobian spectrum by training the denoiser on perturbed inputs, with perturbations suppressing or amplifying Jacobian responses. On ImageNet, we test whether directly modifying the Jacobian spectral properties leads to improved generations. Our findings suggest that denoisers benefit from both strengthening responses along data-relevant principal eigen-directions and suppressing the noisy, data-irrelevant ones. This establishes the denoiser Jacobian as a useful tool for identifying differences between generative denoising models.