Quickest Change Detection with Diffusion-Integrated Scores

2026-10-08

AI Summary

This study addresses the challenge of change-point detection with the classical CUSUM procedure under finite-sample regimes, where likelihood ratios are often intractable. To overcome this limitation, we propose DI-SCUSUM, a novel training-free diffusion-integrated score framework. By smoothing densities with Gaussian noise and combining Hyvärinen scores with importance-weighted recursion, the method achieves efficient detection without requiring trained networks or known distributions. Theoretically, we prove that its increments are proportional to the Kullback–Leibler divergence. Empirical evaluations on both simulated data and the MNIST dataset demonstrate that DI-SCUSUM reduces detection delay by approximately 91% compared to baseline methods, achieving performance closely approaching that of an ideal likelihood ratio detector.

Abstract

Classical CUSUM relies on the log-likelihood ratio of the underlying distributions, which cannot generally be computed from finite pre- and post-change samples alone. We propose diffusion-integrated score CUSUM (DI-SCUSUM), a training-free detector. We add Gaussian noise to the samples to form two smooth density estimates and calculate their Hyvärinen scores exactly, without training a score network. For each incoming observation, we sample a diffusion time, perturb the observation, and use the importance-weighted score difference as an increment in the DI-SCUSUM recursion. Under the assumption that observations follow the fixed empirical distributions, the post-change mean increment is proportional to the Kullback-Leibler (KL) divergence from the smoothed post-change to the smoothed pre-change empirical distribution. We establish exponential false-alarm scaling and a first-order delay bound that, for a fixed threshold and increment scaling, is inversely proportional to the KL divergence. In the calibrated anisotropic Gaussian simulation, DI-SCUSUM nearly matches likelihood-ratio CUSUM and reduces the measured detection delay by about 91% relative to score-based CUSUM. On MNIST and Oxford-IIIT Pet, DI-SCUSUM also has lower empirical conditional detection delay than SCUSUM at comparable false-alarm levels.

Problem

• Quickest Change Detection
• CUSUM
• Log-likelihood Ratio
• Finite Samples

Innovation

• Quickest Change Detection
• DI-SCUSUM
• Diffusion-Integrated Scores
• Training-free
• Hyvärinen Score