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Learning Single Index Models with Diffusion Priors

Anqi Tang, Youming Chen, Shuchen Xue, Zhaoqiang Liu

TL;DR

This work tackles the problem of recovering a high-dimensional signal from nonlinear, possibly discontinuous measurements in a semi-parametric single index model by leveraging diffusion priors. It introduces SIM-DMIS, a method that uses one round of unconditional diffusion model sampling plus partial inversion to reconstruct the signal without knowing the link function $f$. The authors provide theoretical guarantees under mild assumptions and demonstrate substantial reconstruction quality improvements over state-of-the-art baselines while dramatically reducing neural function evaluations across FFHQ, ImageNet, and CIFAR-10 datasets. The approach offers a practical, efficient framework for nonlinear signal recovery with diffusion priors and broad potential impact in imaging and related inverse problems.

Abstract

Diffusion models (DMs) have demonstrated remarkable ability to generate diverse and high-quality images by efficiently modeling complex data distributions. They have also been explored as powerful generative priors for signal recovery, resulting in a substantial improvement in the quality of reconstructed signals. However, existing research on signal recovery with diffusion models either focuses on specific reconstruction problems or is unable to handle nonlinear measurement models with discontinuous or unknown link functions. In this work, we focus on using DMs to achieve accurate recovery from semi-parametric single index models, which encompass a variety of popular nonlinear models that may have {\em discontinuous} and {\em unknown} link functions. We propose an efficient reconstruction method that only requires one round of unconditional sampling and (partial) inversion of DMs. Theoretical analysis on the effectiveness of the proposed methods has been established under appropriate conditions. We perform numerical experiments on image datasets for different nonlinear measurement models. We observe that compared to competing methods, our approach can yield more accurate reconstructions while utilizing significantly fewer neural function evaluations.

Learning Single Index Models with Diffusion Priors

TL;DR

This work tackles the problem of recovering a high-dimensional signal from nonlinear, possibly discontinuous measurements in a semi-parametric single index model by leveraging diffusion priors. It introduces SIM-DMIS, a method that uses one round of unconditional diffusion model sampling plus partial inversion to reconstruct the signal without knowing the link function . The authors provide theoretical guarantees under mild assumptions and demonstrate substantial reconstruction quality improvements over state-of-the-art baselines while dramatically reducing neural function evaluations across FFHQ, ImageNet, and CIFAR-10 datasets. The approach offers a practical, efficient framework for nonlinear signal recovery with diffusion priors and broad potential impact in imaging and related inverse problems.

Abstract

Diffusion models (DMs) have demonstrated remarkable ability to generate diverse and high-quality images by efficiently modeling complex data distributions. They have also been explored as powerful generative priors for signal recovery, resulting in a substantial improvement in the quality of reconstructed signals. However, existing research on signal recovery with diffusion models either focuses on specific reconstruction problems or is unable to handle nonlinear measurement models with discontinuous or unknown link functions. In this work, we focus on using DMs to achieve accurate recovery from semi-parametric single index models, which encompass a variety of popular nonlinear models that may have {\em discontinuous} and {\em unknown} link functions. We propose an efficient reconstruction method that only requires one round of unconditional sampling and (partial) inversion of DMs. Theoretical analysis on the effectiveness of the proposed methods has been established under appropriate conditions. We perform numerical experiments on image datasets for different nonlinear measurement models. We observe that compared to competing methods, our approach can yield more accurate reconstructions while utilizing significantly fewer neural function evaluations.

Paper Structure

This paper contains 31 sections, 6 theorems, 46 equations, 9 figures, 10 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related Work
  3. Signal recovery with diffusion models:
  4. Contributions
  5. Notation
  6. Preliminaries for Diffusion Models
  7. The Sampling Process of Diffusion Models
  8. The Inversion of Diffusion Models
  9. Setup and Approaches
  10. Theoretical Analysis
  11. Experiments
  12. Results for FFHQ (256$\times$256)
  13. Results for ImageNet (256$\times$256)
  14. Conclusion
  15. Extended Related Work
  16. ...and 16 more sections

Key Result

Lemma 1

If $\bm{\epsilon} \sim \mathcal{N}(\bm{0},\mathbf{I}_n)$, then we have that the following holds with high probabilityHere and in the rest of the paper, a statement is said to hold with high probability (w.h.p.) if it holds with probability at least $0.99$. where $C$ is a sufficiently large positive constant.

Figures (9)

  • Figure 1: An illustration of our three approaches. For SIM-DMS, we only perform the sampling from $t^*$ to $\epsilon$. For SIM-DMFIS, we perform the full inversion and sampling procedures. For SIM-DMIS, we first perform the inversion from $t^*$ to $T$, and then perform the sampling from $T$ to $\epsilon$.
  • Figure 2: Examples of 1-bit reconstructed images for FFHQ.
  • Figure 3: Examples of cubic reconstructed images for FFHQ.
  • Figure 4: Examples of 1-bit reconstructed images for ImageNet.
  • Figure 5: Examples of cubic reconstructed images for ImageNet.
  • ...and 4 more figures

Theorems & Definitions (9)

  • Remark 1
  • Lemma 1
  • Lemma 2
  • Remark 2
  • Lemma 3
  • Theorem 3
  • Lemma 4
  • Lemma 5
  • proof