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On the Asymptotic Mean Square Error Optimality of Diffusion Models

Benedikt Fesl, Benedikt Böck, Florian Strasser, Michael Baur, Michael Joham, Wolfgang Utschick

TL;DR

This work addresses the lack of theoretical understanding of diffusion models for mean square error (MSE) optimal denoising by proposing a fast, deterministic DM-based denoiser that forwards only stepwise conditional means and initializes at a timestep $\hat{t}$ chosen from the observation's SNR. It establishes a rigorous connection to the Bayesian CME, deriving a new Lipschitz constant dependent solely on DM hyperparameters and proving polynomial-time convergence to the CME under mild conditions, without requiring convergence to the prior. The analysis decomposes errors into prior-convergence and denoising components and provides bounds that hold even under stepwise mis-specification, showing asymptotic vanishing error as $T$ grows. A complementary perspective reveals that diffusion models inherently fuse a strong denoiser with a generative model, whose stochastic re-sampling can be switched on or off to toggle between denoising and generation. Extensive experiments on synthetic and real datasets (including MNIST, Fashion-MNIST, and Librispeech) corroborate the theory, demonstrating that the proposed deterministic denoiser tracks the CME closely and is robust to moderate SNR misalignment, while offering substantial speedups over stochastic sampling.

Abstract

Diffusion models (DMs) as generative priors have recently shown great potential for denoising tasks but lack theoretical understanding with respect to their mean square error (MSE) optimality. This paper proposes a novel denoising strategy inspired by the structure of the MSE-optimal conditional mean estimator (CME). The resulting DM-based denoiser can be conveniently employed using a pre-trained DM, being particularly fast by truncating reverse diffusion steps and not requiring stochastic re-sampling. We present a comprehensive (non-)asymptotic optimality analysis of the proposed diffusion-based denoiser, demonstrating polynomial-time convergence to the CME under mild conditions. Our analysis also derives a novel Lipschitz constant that depends solely on the DM's hyperparameters. Further, we offer a new perspective on DMs, showing that they inherently combine an asymptotically optimal denoiser with a powerful generator, modifiable by switching re-sampling in the reverse process on or off. The theoretical findings are thoroughly validated with experiments based on various benchmark datasets

On the Asymptotic Mean Square Error Optimality of Diffusion Models

TL;DR

This work addresses the lack of theoretical understanding of diffusion models for mean square error (MSE) optimal denoising by proposing a fast, deterministic DM-based denoiser that forwards only stepwise conditional means and initializes at a timestep chosen from the observation's SNR. It establishes a rigorous connection to the Bayesian CME, deriving a new Lipschitz constant dependent solely on DM hyperparameters and proving polynomial-time convergence to the CME under mild conditions, without requiring convergence to the prior. The analysis decomposes errors into prior-convergence and denoising components and provides bounds that hold even under stepwise mis-specification, showing asymptotic vanishing error as grows. A complementary perspective reveals that diffusion models inherently fuse a strong denoiser with a generative model, whose stochastic re-sampling can be switched on or off to toggle between denoising and generation. Extensive experiments on synthetic and real datasets (including MNIST, Fashion-MNIST, and Librispeech) corroborate the theory, demonstrating that the proposed deterministic denoiser tracks the CME closely and is robust to moderate SNR misalignment, while offering substantial speedups over stochastic sampling.

Abstract

Diffusion models (DMs) as generative priors have recently shown great potential for denoising tasks but lack theoretical understanding with respect to their mean square error (MSE) optimality. This paper proposes a novel denoising strategy inspired by the structure of the MSE-optimal conditional mean estimator (CME). The resulting DM-based denoiser can be conveniently employed using a pre-trained DM, being particularly fast by truncating reverse diffusion steps and not requiring stochastic re-sampling. We present a comprehensive (non-)asymptotic optimality analysis of the proposed diffusion-based denoiser, demonstrating polynomial-time convergence to the CME under mild conditions. Our analysis also derives a novel Lipschitz constant that depends solely on the DM's hyperparameters. Further, we offer a new perspective on DMs, showing that they inherently combine an asymptotically optimal denoiser with a powerful generator, modifiable by switching re-sampling in the reverse process on or off. The theoretical findings are thoroughly validated with experiments based on various benchmark datasets
Paper Structure (28 sections, 6 theorems, 60 equations, 11 figures, 2 tables, 2 algorithms)

This paper contains 28 sections, 6 theorems, 60 equations, 11 figures, 2 tables, 2 algorithms.

Table of Contents

  1. INTRODUCTION
  2. RELATED WORK
  3. PRELIMINARIES
  4. MSE-OPTIMAL DENOISING
  5. Discussion of the DM-based Denoiser
  6. A Novel Lipschitz Constant
  7. Analysis under Prior Convergence
  8. Analysis under Stepwise Error Bound
  9. Denoising to Generation via Re-sampling
  10. EXPERIMENTS
  11. CONCLUSION
  12. Training of the Diffusion Model
  13. Proof of Lemma \ref{['lemma:lipschitz']}
  14. Lipschitz Constant for Noise Estimation
  15. Connection of the DM Denoiser's Lipschitz Constant to the SNR
  16. ...and 13 more sections

Key Result

Theorem 1.1

Let ${\bm{y}} = {\bm{x}} + {\bm{n}}\in \mathbb{R}^N$ be a noisy observation with awgn ${\bm{n}}$ and the stepwise denoising error of the dm's reverse process be bounded by $\Delta$. Then, the distance of the proposed denoiser $f_{{\bm{\theta}}}({\bm{y}})$ that utilizes a pre-trained dm with $T$ time with $\gamma >0$ and ${\hat{t}} < T$ being the number of inference steps depending on the observati

Figures (11)

  • Figure 1: Markov chain of the dm’s full reverse process with $T$ steps and visualization of the proposed denoising procedure, where dm steps $t>{\hat{t}}$ (shaded in gray) are omitted for the estimation.
  • Figure 2: Evaluation of the randomgmm with $N=64$ dimensions (top row) and a pre-trained gmm based on MNIST data (bottom row) with $K=128$ components as ground-truth distribution.
  • Figure 3: Comparison of the dm (solid) with the cme (dashed) after each timestep $t$ with $T=300$ for the random (left) and pre-trainedgmm based on MNIST (right) with $K=128$ components.
  • Figure 4: Evaluation of the pre-trainedgmm based on Fashion-MNIST with $K=128$ components.
  • Figure 5: Evaluation of the pre-trainedgmm based on Librispeech with $K=128$ components.
  • ...and 6 more figures

Theorems & Definitions (6)

  • Theorem 1.1: Main result (informal)
  • Lemma 4.1
  • Theorem 4.2
  • Theorem 4.4: Main Result
  • Corollary 4.6
  • Proposition H.1