ABC: Any-Subset Autoregression via Non-Markovian Diffusion Bridges in Continuous Time and Space

2026-04-30

AI Summary

This work addresses the challenge of generating conditional stochastic processes in continuous spatiotemporal settings from arbitrary observation subsets—such as irregularly sampled data or future frames—by proposing an autoregressive generative framework based on non-Markovian diffusion bridges. The method unifies physical time and state evolution within a single continuous stochastic differential equation (SDE), innovatively initializing from neighboring states, injecting noise proportionally to temporal intervals, and explicitly embedding time into the SDE dynamics. The SDE is derived via path-space measure transformation, and training is performed using a path- and time-dependent denoising score matching algorithm. Empirical evaluations on video generation and weather forecasting demonstrate significant improvements over existing approaches, particularly under low-step sampling and irregular conditioning scenarios.

Abstract

Generating continuous-time, continuous-space stochastic processes (e.g., videos, weather forecasts) conditioned on partial observations (e.g., first and last frames) is a fundamental challenge. Existing approaches, (e.g., diffusion models), suffer from key limitations: (1) noise-to-data evolution fails to capture structural similarity between states close in physical time and has unstable integration in low-step regimes; (2) random noise injected is insensitive to the physical process's time elapsed, resulting in incorrect dynamics; (3) they overlook conditioning on arbitrary subsets of states (e.g., irregularly sampled timesteps, future observations). We propose ABC: Any-Subset Autoregressive Models via Non-Markovian Diffusion Bridges in Continuous Time and Space. Crucially, we model the process with one continual SDE whose time variable and intermediate states track the real time and process states. This has provable advantages: (1) the starting point for generating future states is the already-close previous state, rather than uninformative noise; (2) random noise injection scales with physical time elapsed, encouraging physically plausible dynamics with similar time-adjacent states. We derive SDE dynamics via changes-of-measure on path space, yielding another advantage: (3) path-dependent conditioning on arbitrary subsets of the state history and/or future. To learn these dynamics, we derive a path- and time-dependent extension of denoising score matching. Our experiments show ABC's superiority to competing methods on multiple domains, including video generation and weather forecasting.

Problem

• continuous-time stochastic processes
• partial observation conditioning
• arbitrary subset conditioning
• non-Markovian dynamics
• diffusion bridges

Innovation

• Non-Markovian Diffusion Bridges
• Continuous-Time Stochastic Processes
• Path-Dependent Conditioning
• Autoregressive Modeling
• Denoising Score Matching