# Improving Diffusion Generative Models via Truncated Karhunen--Lo`eve Expansion

arXiv: https://arxiv.org/abs/2503.17657 · HTML: https://mlsanity.com/paper/2503.17657/

## 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 27 of its 42 references that are in the map.
- First posted on arXiv 22 Mar 2025.

## Placed in

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

## Builds on (most cited first, 27 of 27)

- [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)
- [LoRA: Low-Rank Adaptation of Large Language Models](https://mlsanity.com/paper/2106.09685/) (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)
- [Classifier-Free Diffusion Guidance](https://mlsanity.com/paper/2207.12598/) (2022)
- [Flow Straight and Fast: Learning to Generate and Transfer Data with Rectified Flow](https://mlsanity.com/paper/2209.03003/) (2022)
- [Deep Learning Face Attributes in the Wild](https://mlsanity.com/paper/1411.7766/) (2014)
- [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)
- [Consistency Models](https://mlsanity.com/paper/2303.01469/) (2023)
- [DPM-Solver++: Fast Solver for Guided Sampling of Diffusion Probabilistic Models](https://mlsanity.com/paper/2211.01095/) (2022)
- [Analytic-DPM: an Analytic Estimate of the Optimal Reverse Variance in Diffusion Probabilistic Models](https://mlsanity.com/paper/2201.06503/) (2022)
- [Common Diffusion Noise Schedules and Sample Steps are Flawed](https://mlsanity.com/paper/2305.08891/) (2023)
- [Cold Diffusion: Inverting Arbitrary Image Transforms Without Noise](https://mlsanity.com/paper/2208.09392/) (2022)
- [Efficient Diffusion Training via Min-SNR Weighting Strategy](https://mlsanity.com/paper/2303.09556/) (2023)
- [SNIPS: Solving Noisy Inverse Problems Stochastically](https://mlsanity.com/paper/2105.14951/) (2021)
- [Improving Diffusion Inverse Problem Solving with Decoupled Noise Annealing](https://mlsanity.com/paper/2407.01521/) (2024)
- [Elucidating the Exposure Bias in Diffusion Models](https://mlsanity.com/paper/2308.15321/) (2023)
- [Align Your Steps: Optimizing Sampling Schedules in Diffusion Models](https://mlsanity.com/paper/2404.14507/) (2024)
- [Blurring Diffusion Models](https://mlsanity.com/paper/2209.05557/) (2022)
- [Fast ODE-based Sampling for Diffusion Models in Around 5 Steps](https://mlsanity.com/paper/2312.00094/) (2023)
- [The Unreasonable Effectiveness of Gaussian Score Approximation for Diffusion Models and its Applications](https://mlsanity.com/paper/2412.09726/) (2024)
- [Exploring the Optimal Choice for Generative Processes in Diffusion Models: Ordinary vs Stochastic Differential Equations](https://mlsanity.com/paper/2306.02063/) (2023)
- [Generative Fractional Diffusion Models](https://mlsanity.com/paper/2310.17638/) (2023)
- [From Score Matching to Diffusion: A Fine-Grained Error Analysis in the Gaussian Setting](https://mlsanity.com/paper/2503.11615/) (2025)
- [PFDiff: Training-Free Acceleration of Diffusion Models Combining Past and Future Scores](https://mlsanity.com/paper/2408.08822/) (2024)

## Abstract (arXiv)

Pretrained diffusion models exhibit a well-known training-sampling mismatch, often attributed to exposure bias and related distribution-shift effects. We provide a quantitative interpretation of this phenomenon through the notion of an effective noise level: empirically, a pretrained denoiser behaves as if trained at a noise level slightly below the nominal schedule. Motivated by this observation, we introduce a training-free sampling strategy based on truncating the Karhunen--Lo`eve (KL) expansion of the Brownian motion driving the forward stochastic differential equation. Truncation yields a finite-dimensional forward process with a reduced noise level that can be adjusted independently of the time discretization. We prove uniform convergence of the truncated process to the original diffusion. To explain the resulting behaviour, we analyse a toy model in which the denoiser is exact but operates at a reduced effective noise level. The analysis predicts a non-monotone response to the sampling noise with a unique interior optimum, located by a one-dimensional sweep over the truncation order. We implement the approach through corresponding truncated reverse-time and probability-flow equations, without modifying the network architecture. Across CIFAR-10, CelebA, ImageNet, and latent-space Stable Diffusion, the truncation order consistently reveals a sweet spot, improving pretrained models in nearly all tested configurations. Training from scratch at a matched truncation order makes that order the network's own sweet spot, accelerates convergence by about 2.8×, and lowers the generation error on CIFAR-10 (Fr'echet Inception Distance 7.14 to 5.47 at matched epochs; best checkpoint 6.74 to 5.23). A L'evy--Ciesielski comparison confirms that finite expansion is broadly beneficial.