# Self-Consistent Generative Paths via Admissible Random Variational Transport

arXiv: https://arxiv.org/abs/2606.08953 · HTML: https://mlsanity.com/paper/2606.08953/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Placed in Flow matching, time series and science, one of 14 directions in Diffusion models.
- Builds on 24 of its 35 references that are in the map.
- First posted on arXiv 8 Jun 2026.

## Placed in

- Diffusion models / [Flow matching, time series and science](https://mlsanity.com/diffusion/flow-matching-time-series-and-science/)

## Builds on (most cited first, 24 of 24)

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022)
- [Flow Straight and Fast: Learning to Generate and Transfer Data with Rectified Flow](https://mlsanity.com/paper/2209.03003/) (2022)
- [Sinkhorn Distances: Lightspeed Computation of Optimal Transportation Distances](https://mlsanity.com/paper/1306.0895/) (2013)
- [DPM-Solver: A Fast ODE Solver for Diffusion Probabilistic Model Sampling in Around 10 Steps](https://mlsanity.com/paper/2206.00927/) (2022)
- [Consistency Models](https://mlsanity.com/paper/2303.01469/) (2023)
- [One-step Diffusion with Distribution Matching Distillation](https://mlsanity.com/paper/2311.18828/) (2023)
- [Stochastic Interpolants: A Unifying Framework for Flows and Diffusions](https://mlsanity.com/paper/2303.08797/) (2023)
- [Improving and generalizing flow-based generative models with minibatch optimal transport](https://mlsanity.com/paper/2302.00482/) (2023)
- [Which Training Methods for GANs do actually Converge?](https://mlsanity.com/paper/1801.04406/) (2018)
- [Mean Flows for One-step Generative Modeling](https://mlsanity.com/paper/2505.13447/) (2025)
- [Fast Sampling of Diffusion Models with Exponential Integrator](https://mlsanity.com/paper/2204.13902/) (2022)
- [One Step Diffusion via Shortcut Models](https://mlsanity.com/paper/2410.12557/) (2024)
- [Consistency Models Made Easy](https://mlsanity.com/paper/2406.14548/) (2024)
- [SANA-Sprint: One-Step Diffusion with Continuous-Time Consistency Distillation](https://mlsanity.com/paper/2503.09641/) (2025)
- [Generator Matching: Generative modeling with arbitrary Markov processes](https://mlsanity.com/paper/2410.20587/) (2024)
- [Deep Equilibrium Approaches to Diffusion Models](https://mlsanity.com/paper/2210.12867/) (2022)
- [Fixed Point Diffusion Models](https://mlsanity.com/paper/2401.08741/) (2024)
- [Wasserstein proximal operators describe score-based generative models and resolve memorization](https://mlsanity.com/paper/2402.06162/) (2024)
- [A Fixed-Point Approach for Causal Generative Modeling](https://mlsanity.com/paper/2404.06969/) (2024)
- [ERDDCI: Exact Reversible Diffusion via Dual-Chain Inversion for High-Quality Image Editing](https://mlsanity.com/paper/2410.14247/) (2024)
- [Straighten Viscous Rectified Flow via Noise Optimization](https://mlsanity.com/paper/2507.10218/) (2025)
- [FastFlow: Accelerating The Generative Flow Matching Models with Bandit Inference](https://mlsanity.com/paper/2602.11105/) (2026)

## Abstract (arXiv)

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.