# Sampling in Unit Time with Kernel Fisher-Rao Flow

arXiv: https://arxiv.org/abs/2401.03892 · HTML: https://mlsanity.com/paper/2401.03892/

## Key facts

- Cited by 27 papers in the map (data through 14 Oct 2026).
- Builds on 35 of its 82 references that are in the map.
- First posted on arXiv 8 Jan 2024.

## Builds on (most cited first, 30 of 35)

- [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)
- [Variational Inference with Normalizing Flows](https://mlsanity.com/paper/1505.05770/) (2015)
- [Normalizing Flows for Probabilistic Modeling and Inference](https://mlsanity.com/paper/1912.02762/) (2019)
- [Stochastic Interpolants: A Unifying Framework for Flows and Diffusions](https://mlsanity.com/paper/2303.08797/) (2023)
- [Diffusion Schr"odinger Bridge with Applications to Score-Based Generative Modeling](https://mlsanity.com/paper/2106.01357/) (2021)
- [Stein Variational Gradient Descent: A General Purpose Bayesian Inference Algorithm](https://mlsanity.com/paper/1608.04471/) (2016)
- [Denoising Diffusion Samplers](https://mlsanity.com/paper/2302.13834/) (2023)
- [Measuring Sample Quality with Kernels](https://mlsanity.com/paper/1703.01717/) (2017)
- [Annealed Flow Transport Monte Carlo](https://mlsanity.com/paper/2102.07501/) (2021)
- [On the geometry of Stein variational gradient descent](https://mlsanity.com/paper/1912.00894/) (2019)
- [Sum-of-Squares Polynomial Flow](https://mlsanity.com/paper/1905.02325/) (2019)
- [Supervised Training of Conditional Monge Maps](https://mlsanity.com/paper/2206.14262/) (2022)
- [Diffusion Generative Flow Samplers: Improving learning signals through partial trajectory optimization](https://mlsanity.com/paper/2310.02679/) (2023)
- [Coupling techniques for nonlinear ensemble filtering](https://mlsanity.com/paper/1907.00389/) (2019)
- [Inference via low-dimensional couplings](https://mlsanity.com/paper/1703.06131/) (2017)
- [Conditional Sampling with Monotone GANs: from Generative Models to Likelihood-Free Inference](https://mlsanity.com/paper/2006.06755/) (2020)
- [On the representation and learning of monotone triangular transport maps](https://mlsanity.com/paper/2009.10303/) (2020)
- [Sliced Iterative Normalizing Flows](https://mlsanity.com/paper/2007.00674/) (2020)
- [Projected Stein Variational Newton: A Fast and Scalable Bayesian Inference Method in High Dimensions](https://mlsanity.com/paper/1901.08659/) (2019)
- [Gradient Flows for Sampling: Mean-Field Models, Gaussian Approximations and Affine Invariance](https://mlsanity.com/paper/2302.11024/) (2023)
- [An Explicit Expansion of the Kullback-Leibler Divergence along its Fisher-Rao Gradient Flow](https://mlsanity.com/paper/2302.12229/) (2023)
- [A connection between Tempering and Entropic Mirror Descent](https://mlsanity.com/paper/2310.11914/) (2023)
- [Adaptive Importance Sampling meets Mirror Descent: a Bias-variance tradeoff](https://mlsanity.com/paper/2110.15590/) (2021)
- [An Optimal Transport Formulation of Bayes' Law for Nonlinear Filtering Algorithms](https://mlsanity.com/paper/2203.11869/) (2022)
- [MixFlows: principled variational inference via mixed flows](https://mlsanity.com/paper/2205.07475/) (2022)
- [Poisson's equation in nonlinear filtering](https://mlsanity.com/paper/1412.5845/) (2014)
- [Scalable Bayesian transport maps for high-dimensional non-Gaussian spatial fields](https://mlsanity.com/paper/2108.04211/) (2021)
- [Ensemble transport smoothing. Part II: Nonlinear updates](https://mlsanity.com/paper/2210.17435/) (2022)

## Built on by (most cited first, 27 of 27)

- [Feynman-Kac Correctors in Diffusion: Annealing, Guidance, and Product of Experts](https://mlsanity.com/paper/2503.02819/) (2025)
- [Solving Inverse Problems via Diffusion-Based Priors: An Approximation-Free Ensemble Sampling Approach](https://mlsanity.com/paper/2506.03979/) (2025)
- [Fisher-Rao Gradient Flow: Geodesic Convexity and Functional Inequalities](https://mlsanity.com/paper/2407.15693/) (2024)
- [Weighted quantization using MMD: From mean field to mean shift via gradient flows](https://mlsanity.com/paper/2502.10600/) (2025)
- [Stein transport for Bayesian inference](https://mlsanity.com/paper/2409.01464/) (2024)
- [Annealing Flow Generative Models Towards Sampling High-Dimensional and Multi-Modal Distributions](https://mlsanity.com/paper/2409.20547/) (2024)
- [Provable Convergence and Limitations of Geometric Tempering for Langevin Dynamics](https://mlsanity.com/paper/2410.09697/) (2024)
- [Efficient, Multimodal, and Derivative-Free Bayesian Inference With Fisher-Rao Gradient Flows](https://mlsanity.com/paper/2406.17263/) (2024)
- [Path-Guided Particle-based Sampling](https://mlsanity.com/paper/2412.03312/) (2024)
- [DICE: Discrete inverse continuity equation for learning population dynamics](https://mlsanity.com/paper/2507.05107/) (2025)
- [Kernel Approximation of Fisher-Rao Gradient Flows](https://mlsanity.com/paper/2410.20622/) (2024)
- [Telegrapher's Generative Model via Kac Flows](https://mlsanity.com/paper/2506.20641/) (2025)
- [Sequential Monte Carlo approximations of Wasserstein--Fisher--Rao gradient flows](https://mlsanity.com/paper/2506.05905/) (2025)
- [Ensemble-Based Annealed Importance Sampling](https://mlsanity.com/paper/2401.15645/) (2024)
- [Inclusive KL Gradient Flows: Otto-Wasserstein, Fisher-Rao-Gaussian, and Local-Estimator Dynamics](https://mlsanity.com/paper/2411.00214/) (2024)
- [Connections between sequential Bayesian inference and evolutionary dynamics](https://mlsanity.com/paper/2411.16366/) (2024)
- [Score-based deterministic density sampling](https://mlsanity.com/paper/2504.18130/) (2025)
- [A note on the unique properties of the Kullback--Leibler divergence for sampling via gradient flows](https://mlsanity.com/paper/2507.04330/) (2025)
- [An operator splitting analysis of Wasserstein--Fisher--Rao gradient flows](https://mlsanity.com/paper/2511.18060/) (2025)
- [To discretize continually: Mean shift interacting particle systems for Bayesian inference](https://mlsanity.com/paper/2605.14142/) (2026)
- [Sequential transport maps using SoS density estimation and α-divergences](https://mlsanity.com/paper/2402.17943/) (2024)
- [Learning Paths for Dynamic Measure Transport: A Control Perspective](https://mlsanity.com/paper/2511.03797/) (2025)
- [Subspace Langevin Monte Carlo](https://mlsanity.com/paper/2412.13928/) (2024)
- [Guiding Time-Varying Generative Models with Natural Gradients on Exponential Family Manifold](https://mlsanity.com/paper/2502.07650/) (2025)
- [Generative Modeling via Kernelized Stochastic Interpolants](https://mlsanity.com/paper/2602.20070/) (2026)
- [Preservation of Log-Concavity and Convergence of Wasserstein-Fisher-Rao Gradient Flows](https://mlsanity.com/paper/2609.18118/) (2026)
- [Interacting particle guidance for sampling reward-tilted generative priors](https://mlsanity.com/paper/2609.37227/) (2026)

## Abstract (arXiv)

We introduce a new mean-field ODE and corresponding interacting particle systems (IPS) for sampling from an unnormalized target density. The IPS are gradient-free, available in closed form, and only require the ability to sample from a reference density and compute the (unnormalized) target-to-reference density ratio. The mean-field ODE is obtained by solving a Poisson equation for a velocity field that transports samples along the geometric mixture of the two densities, which is the path of a particular Fisher-Rao gradient flow. We employ a RKHS ansatz for the velocity field, which makes the Poisson equation tractable and enables discretization of the resulting mean-field ODE over finite samples. The mean-field ODE can be additionally be derived from a discrete-time perspective as the limit of successive linearizations of the Monge-Amp`ere equations within a framework known as sample-driven optimal transport. We introduce a stochastic variant of our approach and demonstrate empirically that our IPS can produce high-quality samples from varied target distributions, outperforming comparable gradient-free particle systems and competitive with gradient-based alternatives.