Self-Consistent Generative Paths via Admissible Random Variational Transport
2026-06-08 • Machine Learning
Machine Learning
AI summaryⓘ
The authors explore when generated probability paths in modern generative models are self-consistent, meaning they don't contradict themselves after small corrections. They introduce a mathematical framework that measures how close a generated path is to being self-consistent using a concept called the random fixed-point path residual (R-FPR). Their theory applies broadly to many model types like diffusion models, VAEs, GANs, and autoregressive generators and helps diagnose and improve model performance by turning endpoint matching into a test of path consistency.
Generative modelsProbability pathsDiffusion modelsVariational transportOptimal transportRandom fixed pointVAEsGANsAutoregressive modelsEndpoint matching
Authors
Lei Luo, Yingzhen Zhang, Jian Yang
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.