Learnability of the FlippedSNR Forward Process in Diffusion Models

Determine whether diffusion models trained with the FlippedSNR forward process—defined in Fourier space by setting the per-frequency noise covariance Σii = Ci/Cd−i so that the Signal-to-Noise Ratio (SNR) of frequency i equals the SNR of frequency d−i and thereby inverts the DDPM low-to-high frequency hierarchy—can learn a reverse process that accurately approximates the target data distribution; equivalently, prove or refute that such a flipped-frequency forward process is learnable.

Background

The paper proposes analyzing diffusion models in Fourier space, highlighting that standard DDPM noising corrupts high frequencies faster and earlier, which induces a low-to-high frequency generation hierarchy in the reverse process. To paper the role of this hierarchy, the authors introduce alternate forward processes defined via per-frequency SNR, including EqualSNR (equal SNR across frequencies) and FlippedSNR (inverting the DDPM hierarchy so that low frequencies are noised first).

While EqualSNR trains successfully and performs competitively, multiple attempts to train FlippedSNR failed to produce a learned reverse process that approximates the data distribution. Despite this empirical difficulty, the authors explicitly note that they cannot prove impossibility, leaving open whether a diffusion model with the FlippedSNR forward process can in principle be trained to match the data distribution. Resolving this would clarify whether the DDPM low-to-high frequency hierarchy is necessary for effective diffusion model training.