Quantize Where the Noise Hides: A Self-Correction Theorem and Water-Filling Bit Allocation for Diffusion Sampling

Xiaoyu Li, Zheng Gao, Zhizhou Sha, Jiaojiao Jiang

IEEE/CVF Winter Conference on Applications of Computer Vision (WACV), Algorithms Track, 2027

Abstract

Post-training quantization of diffusion models is an active area of empirical research, with one robust but unexplained regularity: the early, high-noise reverse steps tolerate coarse precision, while the late, low-noise steps do not. There is no theory of why this happens, and no principled way to spend a fixed bit budget across timesteps. We supply both, from a single principle: the reverse process has an error-correction capacity equal to its own injected noise. Treating the quantization applied at reverse step t as an added error of magnitude ε_t, we prove a self-correction theorem: the quantized and exact sampling laws stay within Wasserstein-2 distance ∑_t s_tε_t (a non-asymptotic bound), where the sensitivity s_t is the factor by which an error at step t is amplified at the output. As long as a step's quantization stays below the noise the schedule already injects at that step, the error is absorbed; and because that noise shrinks toward the final step, the per-step tolerance tightens monotonically, which is exactly the early-coarse / late-fine rule. Given a total budget, the sampling error is then minimized by water-filling the bits across timesteps according to the sensitivities s_t, the classical rate-distortion allocation transplanted to diffusion, with uniform and full precision as its two limiting cases. The result is a principled mixed-precision schedule with a Wasserstein guarantee that is exact on closed-form targets. On a small trained model the schedule beats uniform precision and a PTQD-style SNR-matching heuristic at matched average bit-width; on a pretrained CIFAR-10 DDPM it is the most faithful allocation to the full-precision sampler at every budget and matches the best heuristic's FID within the theory's validity range, with the coarser-budget gap diagnosed as bias-dominated.

BibTeX

@inproceedings{li2027quantizewherenoisehides,
  title     = {Quantize Where the Noise Hides: A Self-Correction Theorem and Water-Filling Bit Allocation for Diffusion Sampling},
  author    = {Li, Xiaoyu and Gao, Zheng and Sha, Zhizhou and Jiang, Jiaojiao},
  booktitle = {IEEE/CVF Winter Conference on Applications of Computer Vision (WACV)},
  note      = {Algorithms Track},
  year      = {2027},
}