Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Beyond Fixed Horizons: A Theoretical Framework for Adaptive Denoising Diffusions

Sören Christensen, Jan Kallsen, Claudia Strauch, Lukas Trottner

TL;DR

The paper develops a novel class of adaptive diffusion models that replace fixed time horizons with random terminal times using Doob's $h$-transform, yielding time-homogeneous forward and backward dynamics and adaptive denoising steps. It provides a theoretical foundation linking $h$-transforms to stochastic control, time-reversal, and optimal stopping, and shows how to learn the backward drift while leveraging a polarity (manifold-like) data structure to terminate generation. The approach enables natural conditioning, flexible unconditional sampling through multiple horizon constructions, and downstream tasks such as conditional data generation, anomaly detection, and class-aware sampling without task-specific redesigns. By unifying random horizons with manifold learning, the framework offers a robust, transfer-learning-friendly foundation for diffusion modeling with interpretable stopping rules and potential for diverse applications. The work also demonstrates how classical noise schedules emerge as radial $h$-transforms, connecting new theory to established diffusion practices and highlighting scalable paths for high-dimensional data.

Abstract

We introduce a new class of generative diffusion models that, unlike conventional denoising diffusion models, achieve a time-homogeneous structure for both the noising and denoising processes, allowing the number of steps to adaptively adjust based on the noise level. This is accomplished by conditioning the forward process using Doob's $h$-transform, which terminates the process at a suitable sampling distribution at a random time. The model is particularly well suited for generating data with lower intrinsic dimensions, as the termination criterion simplifies to a first-hitting rule. A key feature of the model is its adaptability to the target data, enabling a variety of downstream tasks using a pre-trained unconditional generative model. These tasks include natural conditioning through appropriate initialisation of the denoising process and classification of noisy data.

Beyond Fixed Horizons: A Theoretical Framework for Adaptive Denoising Diffusions

TL;DR

The paper develops a novel class of adaptive diffusion models that replace fixed time horizons with random terminal times using Doob's -transform, yielding time-homogeneous forward and backward dynamics and adaptive denoising steps. It provides a theoretical foundation linking -transforms to stochastic control, time-reversal, and optimal stopping, and shows how to learn the backward drift while leveraging a polarity (manifold-like) data structure to terminate generation. The approach enables natural conditioning, flexible unconditional sampling through multiple horizon constructions, and downstream tasks such as conditional data generation, anomaly detection, and class-aware sampling without task-specific redesigns. By unifying random horizons with manifold learning, the framework offers a robust, transfer-learning-friendly foundation for diffusion modeling with interpretable stopping rules and potential for diverse applications. The work also demonstrates how classical noise schedules emerge as radial -transforms, connecting new theory to established diffusion practices and highlighting scalable paths for high-dimensional data.

Abstract

We introduce a new class of generative diffusion models that, unlike conventional denoising diffusion models, achieve a time-homogeneous structure for both the noising and denoising processes, allowing the number of steps to adaptively adjust based on the noise level. This is accomplished by conditioning the forward process using Doob's -transform, which terminates the process at a suitable sampling distribution at a random time. The model is particularly well suited for generating data with lower intrinsic dimensions, as the termination criterion simplifies to a first-hitting rule. A key feature of the model is its adaptability to the target data, enabling a variety of downstream tasks using a pre-trained unconditional generative model. These tasks include natural conditioning through appropriate initialisation of the denoising process and classification of noisy data.

Paper Structure

This paper contains 25 sections, 8 theorems, 72 equations, 7 algorithms.

Table of Contents

  1. Introduction
  2. Related work
  3. Doob's $h$-transform and time-inversion from random times
  4. Doob's $h$-transform for random time horizons
  5. Optimality properties of the $h$-transform
  6. Connection to time-inversion
  7. Key examples
  8. The proposed model
  9. Three possible implementations for unconditional sampling
  10. Polarity hypothesis
  11. Learning the drift
  12. (Implicit) manifold learning
  13. Approach based on separation of manifold and drift estimation
  14. Fully integrated drift and support estimation
  15. Noise-Schedules as a special case of radial $h$-transforms
  16. ...and 10 more sections

Key Result

Proposition 3.1

Theorems & Definitions (17)

  • Proposition 3.1
  • Proposition 3.2
  • Proposition 3.3
  • Proposition 3.4
  • proof
  • Proposition 4.1
  • Lemma 4.2
  • Remark 4.3
  • Proposition 4.4
  • proof
  • ...and 7 more