When does the noise schedule matter? A spectral classification of diffusion training objectives

Diffusion models require a noise schedule, yet for a given objective it is not generally clear before training whether schedule choice should affect trained-model quality. This leaves a diagnostic question open: given a training objective and a dataset, does the objective admit a pre-training spectral criterion for schedule choice? We show that the answer is determined by the structure of the per-step loss. Product-form training objectives admit a Schedule Mismatch Index, a scalar measuring how unevenly a schedule distributes the training signal across timesteps, computable from data-covariance eigenvalues alone, that strictly ranks schedules by proxy-estimator variance; point-evaluation training objectives do not admit the same construction (Theorem 1). This matches the empirical contrast between VLB training, where cosine beats linear by 1.4% on CIFAR-10, and $ϵ$-prediction, where the top three schedules fall within 0.5%. The same framework yields a training-aligned timestep sampler $q_{\mathrm{train}}^{\star }$ that improves CIFAR-10 by 2.61% under VLB, with concordant gains on Fisher–Rao-weighted evaluator and FID. At 100k iterations across CIFAR-10 and ImageNet-32 with U-Net and DiT-S backbones, the improvement persists in all four dataset-architecture settings (1.99%–2.14%), while the closed-form proxy remains indistinguishable from uniform. These results show that schedule and timestep-sampling choices must be analyzed relative to the trained objective, not the data spectrum alone.