An Eulerian Perspective on Straight-Line Sampling

2025-10-13

AI Summary

This work addresses the optimization of flow architectures for generative modeling in dynamic optimal transport, specifically targeting *straight flows*—stochastic processes whose particle accelerations vanish identically and are thus exactly integrable by first-order numerical schemes. Adopting an Eulerian perspective, we model the velocity field via conditional expectations and derive a novel partial differential equation characterizing straight flows: its solutions must satisfy exact balance between conditional acceleration and the divergence of a weighted covariance tensor. We prove that, under affine time interpolation, straight flows exist if and only if the source and target distributions admit a deterministic endpoint coupling. This result provides the first complete necessary and sufficient geometric and probabilistic characterization of zero-acceleration flows. It establishes a rigorous theoretical foundation and constructive design principles for analytically integrable, computationally efficient normalizing flow models.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Machine Learning: Learning with Manifolds
• Search and Optimization: Non-convex Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• User Modeling, Personalization and Recommendation: On-Device user modeling, personalization, and recommendation
• Semantics and Knowledge: Data modeling to support human-machine intelligence, including LLMs agents, intelligent system behavior, explanations, and user-friendly interactions

Abstract

We study dynamic measure transport for generative modeling: specifically, flows induced by stochastic processes that bridge a specified source and target distribution. The conditional expectation of the process' velocity defines an ODE whose flow map achieves the desired transport. We ask emph{which processes produce straight-line flows} -- i.e., flows whose pointwise acceleration vanishes and thus are exactly integrable with a first-order method? We provide a concise PDE characterization of straightness as a balance between conditional acceleration and the divergence of a weighted covariance (Reynolds) tensor. Using this lens, we fully characterize affine-in-time interpolants and show that straightness occurs exactly under deterministic endpoint couplings. We also derive necessary conditions that constrain flow geometry for general processes, offering broad guidance for designing transports that are easier to integrate.

Problem

• Characterizing stochastic processes producing straight-line transport flows
• Providing PDE conditions for vanishing acceleration in generative models
• Establishing constraints on flow geometry for easier numerical integration

Innovation

• Stochastic processes bridge source and target distributions
• PDE characterization balances acceleration and covariance divergence
• Straight-line flows occur under deterministic endpoint couplings