Self-Consistent Generative Paths via Admissible Random Variational Transport

2026-06-07

AI Summary

This work addresses a critical limitation in existing generative models, which typically focus on matching endpoint distributions while neglecting the self-consistency of the generation path. The paper introduces the notion of a “self-consistent generative path,” formalized as a stochastic fixed point amenable to local variational transport corrections, and establishes the first unified theoretical framework for analyzing such paths. By integrating optimal transport, energy functionals, and fixed-point theory through a novel stochastic variational transport operator and a path residual metric, the framework systematically characterizes the path behavior of diverse models—including diffusion models, flow matching, VAEs, and GANs. Theoretically, the existence and attractivity of the stochastic fixed point are proven, and rigorous bounds are derived linking path residuals to generation error, along with results on empirical concentration and operator generalization, yielding a unified diagnostic and optimization criterion applicable across multiple generative model families.

Abstract

Modern generative models often define an entire probability path from a simple prior to the data law, rather than only an endpoint map. Diffusion models follow stochastic denoising paths, flow matching learns transport fields, consistency and distillation methods compress paths into one or a few steps, adversarial models match terminal distributions, and VAEs generate through latent kernels. Existing unifying views mainly describe how such paths are constructed. We study a complementary question: when is a generated probability path self-consistent? We define a self-consistent generative path as a random fixed point of admissible local variational transport corrections. In this framework, a local correction is specified by a random variational transport operator combining a divergence or geometry term, an energy term, and a structural constraint. The framework contains random regularized optimal-transport proximal steps as a structured instance, while also allowing non-OT divergences, latent kernels, adversarial constraints, causal discrete kernels, and terminal one-step maps. The theory yields a random fixed-point path residual (R-FPR), which measures the gap between the actual generated path and an admissible local correction. We prove well-posedness, random fixed-point existence and attraction, non-contractive existence, residual-to-generation error bounds, empirical residual concentration, proxy perturbation bounds, continuous-time limits, and operator-level generalization with model-specific corollaries. The resulting theory turns endpoint matching into path self-consistency testing and provides a residual-control principle for diagnosing failures, regularizing training, and guiding adaptive sampling across diffusion, flow, one-step, VAE, GAN/WGAN, and autoregressive generators.

Problem

• self-consistency
• generative paths
• variational transport
• probability path
• random fixed point

Innovation

• self-consistent generative paths
• random variational transport
• fixed-point path residual
• admissible transport operators
• generative model unification