Variance Path Diffusion: Per-Region Noise Schedules for Inpainting and Motion In-Betweening

Diffusion models corrupt each coordinate on a single, predefined noise schedule, yet most conditional tasks, such as inpainting, local editing, and motion in-betweening, require noise levels that differ across space and time. Existing methods design such region-wise schedules by hand, each rediscovering empirically the constraints a schedule must satisfy. We derive these constraints instead, by modeling the variance as a random variable: Variance Path Diffusion (VPD) models the cumulative noise as a Gamma process pinned in expectation to a nominal schedule, making variance allocation a free design variable. We prove that the natural shared-rate construction cannot meet a sufficient condition for well-behaved training at any finite stochasticity, and repair it with a Time-Varying-Rate (TVR) construction based on Beta thinning, which restores admissible VPD properties while leaving the marginals intact, and show that the randomness of the path is bounded and empirically does not affect quality. What remains is structure. A single allocation parameter $\gamma$, applied at sampling time to one model trained once, spans unconditional generation, editing, and inpainting. VPD covers temporal allocation for motion. VPD achieves the lowest or tied all-joint keyframe error on HumanML3D in-betweening, and sets a new state of the art for inpainting on FFHQ, LSUN Bedroom, and ImageNet.