# Flow Perturbation++: Multi-Step Unbiased Jacobian Estimation for High-Dimensional Boltzmann Sampling

arXiv: https://arxiv.org/abs/2601.21177 · HTML: https://mlsanity.com/paper/2601.21177/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 27 of its 52 references that are in the map.
- First posted on arXiv 29 Jan 2026.

## Builds on (most cited first, 27 of 27)

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- [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)
- [Soft Actor-Critic: Off-Policy Maximum Entropy Deep Reinforcement Learning with a Stochastic Actor](https://mlsanity.com/paper/1801.01290/) (2018)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Variational Inference with Normalizing Flows](https://mlsanity.com/paper/1505.05770/) (2015)
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- [Glow: Generative Flow with Invertible 1x1 Convolutions](https://mlsanity.com/paper/1807.03039/) (2018)
- [Equivariant Diffusion for Molecule Generation in 3D](https://mlsanity.com/paper/2203.17003/) (2022)
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- [GeoDiff: a Geometric Diffusion Model for Molecular Conformation Generation](https://mlsanity.com/paper/2203.02923/) (2022)
- [Invertible Residual Networks](https://mlsanity.com/paper/1811.00995/) (2018)
- [GFlowNet Foundations](https://mlsanity.com/paper/2111.09266/) (2021)
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- [DiT: Self-supervised Pre-training for Document Image Transformer](https://mlsanity.com/paper/2203.02378/) (2022)
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- [Annealed Flow Transport Monte Carlo](https://mlsanity.com/paper/2102.07501/) (2021)
- [Conv-Basis: A New Paradigm for Efficient Attention Inference and Gradient Computation in Transformers](https://mlsanity.com/paper/2405.05219/) (2024)
- [Maximum Entropy Reinforcement Learning via Energy-Based Normalizing Flow](https://mlsanity.com/paper/2405.13629/) (2024)
- [Diffusion Models as Stochastic Quantization in Lattice Field Theory](https://mlsanity.com/paper/2309.17082/) (2023)
- [Soft Policy Gradient Method for Maximum Entropy Deep Reinforcement Learning](https://mlsanity.com/paper/1909.03198/) (2019)
- [S2AC: Energy-Based Reinforcement Learning with Stein Soft Actor Critic](https://mlsanity.com/paper/2405.00987/) (2024)
- [Sampling from Energy-based Policies using Diffusion](https://mlsanity.com/paper/2410.01312/) (2024)
- [Verlet Flows: Exact-Likelihood Integrators for Flow-Based Generative Models](https://mlsanity.com/paper/2405.02805/) (2024)
- [Statistical Distance Based Deterministic Offspring Selection in SMC Methods](https://mlsanity.com/paper/2212.12290/) (2022)

## Abstract (arXiv)

The scalability of continuous normalizing flows (CNFs) for unbiased Boltzmann sampling remains limited in high-dimensional systems due to the cost of Jacobian-determinant evaluation, which requires D backpropagation passes through the flow layers. Existing stochastic Jacobian estimators such as the Hutchinson trace estimator reduce computation but introduce bias, while the recently proposed Flow Perturbation method is unbiased yet suffers from high variance. We present Flow Perturbation++, a variance-reduced extension of Flow Perturbation that discretizes the probability-flow ODE and performs unbiased stepwise Jacobian estimation at each integration step. This multi-step construction retains the unbiasedness of Flow Perturbation while achieves substantially lower estimator variance. Integrated into a Sequential Monte Carlo framework, Flow Perturbation++ achieves significantly improved equilibrium sampling on a 1000D Gaussian Mixture Model and the all-atom Chignolin protein compared with Hutchinson-based and single-step Flow Perturbation baselines.