Limits of Fixed Score Scaling in Diffusion Models under Gaussian Mixtures

Diffusion-based image reconstruction involves a perception–distortion (PD) tradeoff. Fixed score scaling adjusts this tradeoff without retraining by applying the same score multiplier at every noise level. For Gaussian sources, doubling the score in the variance-preserving probability-flow ordinary differential equation recovers the posterior mean and minimizes mean-squared error (MSE). However, this result does not guarantee posterior-mean recovery for non-Gaussian sources. For every nontrivial finite Gaussian mixture model (GMM) with positive weights and positive-definite component covariances, we prove a positive lower bound on excess MSE over all real constant multipliers. The bound extends to deterministic denoising diffusion implicit model (DDIM) sampling on sufficiently fine grids over expanding multiplier intervals. To approximate the posterior mean, we propose Simple Endpoint Correction, which combines two constant-scaling endpoints with coefficients selected on a calibration set. Experiments show lower held-out negative log-likelihood for GMMs than for single Gaussians on the evaluated image latents. Endpoint correction reduces posterior-mean error on synthetic GMMs and image-space MSE for DiffC-SD2.1, ResULIC, and ELIC+DDIM/DSG.