Diffusion models recover accurate mixture weights despite score function insensitivity

2026-07-16

AI Summary

Although diffusion models can capture all modes of multimodal distributions, they struggle to accurately recover the mixture weights of individual modes. This work reveals that score information at intermediate noise levels is critical for precise weight estimation and introduces the Diffusion Score Sensitivity Index (DSSI), which establishes—for the first time—a theoretical connection between the Denoising Score Matching (DSM) loss and parameter estimation error in Gaussian mixture models. Theoretically, the weight estimation error is shown to be of the same order as the DSM loss. Empirically, under common noise schedules, DSSI effectively predicts the performance of weight recovery, and the study further identifies that improper scheduling can lead to mode amplification.

Abstract

Score-based generative models exhibit a puzzling behavior: they often appear to cover all modes of a target multimodal distribution and yet may fail to learn the correct relative mode amplitudes, which can be interpreted as mixture weights. We resolve this apparent paradox by relating the diffusion score matching (DSM) loss to the error in estimating mixture weights from generated samples. We show that, even when the target score is insensitive to mixture weights, generated samples can recover the weights accurately if the scores at intermediate noise levels are informative about the weights. Accordingly, we define the diffusion score sensitivity index (DSSI) as the variation in the DSM loss relative to changes in a parameter. We then show that the DSSI governs the accuracy with which the parameter of the target distribution can be estimated from generated samples. For Gaussian mixtures in arbitrary dimensions, we prove that the mixture weight estimation errors are on the same order as the DSM loss under mild conditions. Empirically, we show the emergence of sensitivity during the noising process of benchmark data distributions under typical noise schedules, and that these sensitivity values predict how well a well-trained model recovers mixture weights. Furthermore, we show that the choice of noise schedule can reduce diffusion sensitivity, leading to mode amplification. Although we focus on mixture weights, the proposed sensitivity framework governs the recovery of any qualitative parameter of the target distribution.

Problem

• diffusion models
• mixture weights
• score-based generative models
• multimodal distribution
• mode amplitudes

Innovation

• diffusion score sensitivity index
• score-based generative models
• mixture weight estimation
• diffusion score matching
• noise schedule