Understanding Intensity Saturation in Diffusion Models through Data Covariance

Diffusion models exhibit approximately linear behavior at moderate to high noise levels, but how this linearity manifests in final generated samples remains less understood. We show that projections of initial noise and generated samples onto the same data covariance eigenvector generally exhibit a more linear relationship for eigenvectors with larger eigenvalues. In natural image datasets, leading eigenvectors with larger eigenvalues represent broad brightness variations. Large-magnitude initial noise projections along these directions can therefore contribute to spatially coherent regions becoming extremely bright or dark, a phenomenon we call *intensity saturation*. Classifier-free guidance (CFG) changes both the type of intensity saturation and the saturated area. Without guidance, saturation is more commonly dominated by either dark or bright regions, while their coexistence becomes more frequent as the CFG weight increases. Overall, increasing the CFG weight increases intensity saturation. However, it can increase or decrease the saturated area of individual samples. Our linear analysis suggests that these changes depend on the class and initial noise projections through changes in coefficient magnitudes along the leading eigenvectors. Motivated by these findings, we propose a method that selectively damps large coefficients along these directions to reduce intensity saturation, thereby improving image quality under guidance.