# Score-based deterministic density sampling

arXiv: https://arxiv.org/abs/2504.18130 · HTML: https://mlsanity.com/paper/2504.18130/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 32 of its 43 references that are in the map.
- First posted on arXiv 25 Apr 2025.

## Builds on (most cited first, 30 of 32)

- [Denoising Diffusion Implicit Models](https://mlsanity.com/paper/2010.02502/) (2020)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Generative Modeling by Estimating Gradients of the Data Distribution](https://mlsanity.com/paper/1907.05600/) (2019)
- [Elucidating the Design Space of Diffusion-Based Generative Models](https://mlsanity.com/paper/2206.00364/) (2022)
- [DPM-Solver: A Fast ODE Solver for Diffusion Probabilistic Model Sampling in Around 10 Steps](https://mlsanity.com/paper/2206.00927/) (2022)
- [Stein Variational Gradient Descent: A General Purpose Bayesian Inference Algorithm](https://mlsanity.com/paper/1608.04471/) (2016)
- [Sampling is as easy as learning the score: theory for diffusion models with minimal data assumptions](https://mlsanity.com/paper/2209.11215/) (2022)
- [Rapid Convergence of the Unadjusted Langevin Algorithm: Isoperimetry Suffices](https://mlsanity.com/paper/1903.08568/) (2019)
- [Stein Variational Gradient Descent as Gradient Flow](https://mlsanity.com/paper/1704.07520/) (2017)
- [The probability flow ODE is provably fast](https://mlsanity.com/paper/2305.11798/) (2023)
- [Sampling as optimization in the space of measures: The Langevin dynamics as a composite optimization problem](https://mlsanity.com/paper/1802.08089/) (2018)
- [User-friendly guarantees for the Langevin Monte Carlo with inaccurate gradient](https://mlsanity.com/paper/1710.00095/) (2017)
- [Interacting particle solutions of Fokker-Planck equations through gradient-log-density estimation](https://mlsanity.com/paper/2006.00702/) (2020)
- [On the geometry of Stein variational gradient descent](https://mlsanity.com/paper/1912.00894/) (2019)
- [Convergence Analysis of Probability Flow ODE for Score-based Generative Models](https://mlsanity.com/paper/2404.09730/) (2024)
- [Message Passing Stein Variational Gradient Descent](https://mlsanity.com/paper/1711.04425/) (2017)
- [Probability flow solution of the Fokker-Planck equation](https://mlsanity.com/paper/2206.04642/) (2022)
- [Learning Interpolations between Boltzmann Densities](https://mlsanity.com/paper/2301.07388/) (2023)
- [Normalizing flow neural networks by JKO scheme](https://mlsanity.com/paper/2212.14424/) (2022)
- [Sampling in Unit Time with Kernel Fisher-Rao Flow](https://mlsanity.com/paper/2401.03892/) (2024)
- [Score-based Transport Modeling for Mean-Field Fokker-Planck Equations](https://mlsanity.com/paper/2305.03729/) (2023)
- [Bounds for the smallest eigenvalue of the NTK for arbitrary spherical data of arbitrary dimension](https://mlsanity.com/paper/2405.14630/) (2024)
- [Neural Sampling from Boltzmann Densities: Fisher-Rao Curves in the Wasserstein Geometry](https://mlsanity.com/paper/2410.03282/) (2024)
- [Measure transport with kernel mean embeddings](https://mlsanity.com/paper/2401.12967/) (2024)
- [Noise-Free Sampling Algorithms via Regularized Wasserstein Proximals](https://mlsanity.com/paper/2308.14945/) (2023)
- [A score-based particle method for homogeneous Landau equation](https://mlsanity.com/paper/2405.05187/) (2024)
- [A Practical Diffusion Path for Sampling](https://mlsanity.com/paper/2406.14040/) (2024)
- [Annealed Stein Variational Gradient Descent for Improved Uncertainty Estimation in Full-Waveform Inversion](https://mlsanity.com/paper/2410.13249/) (2024)
- [On the query complexity of sampling from non-log-concave distributions](https://mlsanity.com/paper/2502.06200/) (2025)
- [Minimum Kernel Discrepancy Estimators](https://mlsanity.com/paper/2210.16357/) (2022)

## Built on by (most cited first, 4 of 4)

- [DiScoFormer: Plug-In Density and Score Estimation with Transformers](https://mlsanity.com/paper/2511.05924/) (2025)
- [Alternating Diffusion for Proximal Sampling with Zeroth Order Queries](https://mlsanity.com/paper/2603.19633/) (2026)
- [A Neural Score-Based Particle Method for the Vlasov-Maxwell-Landau System](https://mlsanity.com/paper/2603.25832/) (2026)
- [Wasserstein Gradient Flows and Forward-Only Diffusion Are Not Enough for Multimodal Sampling](https://mlsanity.com/paper/2610.02081/) (2026)

## Abstract (arXiv)

We propose a deterministic sampling framework using Score-Based Transport Modeling for sampling an unnormalized target density π given only its score nabla log π. Our method approximates the Wasserstein gradient flow on KL(ft|π) by learning the time-varying score nabla log ft on the fly using score matching. While having the same marginal distribution as Langevin dynamics, our method produces smooth deterministic trajectories, resulting in monotone noise-free convergence. We prove that our method dissipates relative entropy at the same rate as the exact gradient flow, provided sufficient training. Numerical experiments validate our theoretical findings: our method converges at the optimal rate, has smooth trajectories, and is often more sample efficient than its stochastic counterpart. Experiments on high-dimensional image data show that our method produces high-quality generations in as few as 15 steps and exhibits natural exploratory behavior. The memory and runtime scale linearly in the sample size.