Title: Provably Efficient High-Order Flow Matching in Pixel Space
Primary area: generative models
Keywords: Generative Model, Flow Matching, Diffusion

Abstract: We introduce high-order PixelFlow (HopeFlow), which is the first cascade flow model that learns both pixel‑space velocity and acceleration fields end‑to‑end, lifting image generation beyond the limitations of purely first‑order supervision. By incorporating second‑order dynamics, HopeFlow aligns mid‑horizon dependencies and high‑curvature regions, yielding markedly smoother, more stable transport trajectories. The model trains directly on raw pixels—no VAE encoder‑decoder is required—and remains computationally affordable. We prove that the HopeFlow model is computable by a $\mathsf{TC}^0$ class of threshold circuits, which operate with constant depth $O(1)$ and a polynomial number of gates $\mathrm{poly}(n)$. Moreover, by replacing exact attention with approximate attention layers, the end‐to‐end HopeFlow inference runs in almost quadratic time.