# Harmonic Path Integral Diffusion

arXiv: https://arxiv.org/abs/2409.15166 · HTML: https://mlsanity.com/paper/2409.15166/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- Builds on 24 of its 46 references that are in the map.
- First posted on arXiv 23 Sep 2024.

## 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)
- [Deep Unsupervised Learning using Nonequilibrium Thermodynamics](https://mlsanity.com/paper/1503.03585/) (2015)
- [Adjoint Matching: Fine-tuning Flow and Diffusion Generative Models with Memoryless Stochastic Optimal Control](https://mlsanity.com/paper/2409.08861/) (2024)
- [Likelihood Training of Schr"odinger Bridge using Forward-Backward SDEs Theory](https://mlsanity.com/paper/2110.11291/) (2021)
- [Path Integral Sampler: a stochastic control approach for sampling](https://mlsanity.com/paper/2111.15141/) (2021)
- [Dynamical Regimes of Diffusion Models](https://mlsanity.com/paper/2402.18491/) (2024)
- [Flow Annealed Importance Sampling Bootstrap](https://mlsanity.com/paper/2208.01893/) (2022)
- [Iterated Denoising Energy Matching for Sampling from Boltzmann Densities](https://mlsanity.com/paper/2402.06121/) (2024)
- [Improved sampling via learned diffusions](https://mlsanity.com/paper/2307.01198/) (2023)
- [Theoretical guarantees for sampling and inference in generative models with latent diffusions](https://mlsanity.com/paper/1903.01608/) (2019)
- [Probabilistic Forecasting with Stochastic Interpolants and F"ollmer Processes](https://mlsanity.com/paper/2403.13724/) (2024)
- [Beyond ELBOs: A Large-Scale Evaluation of Variational Methods for Sampling](https://mlsanity.com/paper/2406.07423/) (2024)
- [Wasserstein Proximal Algorithms for the Schr"odinger Bridge Problem: Density Control with Nonlinear Drift](https://mlsanity.com/paper/1912.01244/) (2019)
- [Improved off-policy training of diffusion samplers](https://mlsanity.com/paper/2402.05098/) (2024)
- [Transferable Boltzmann Generators](https://mlsanity.com/paper/2406.14426/) (2024)
- [Generalized Schr"odinger Bridge Matching](https://mlsanity.com/paper/2310.02233/) (2023)
- [Towards Predicting Equilibrium Distributions for Molecular Systems with Deep Learning](https://mlsanity.com/paper/2306.05445/) (2023)
- [Adaptive importance sampling for control and inference](https://mlsanity.com/paper/1505.01874/) (2015)
- [Schr"odinger Bridge with Quadratic State Cost is Exactly Solvable](https://mlsanity.com/paper/2406.00503/) (2024)
- [U-Turn Diffusion](https://mlsanity.com/paper/2308.07421/) (2023)
- [Space-Time Diffusion Bridge](https://mlsanity.com/paper/2402.08847/) (2024)
- [Weyl Calculus and Exactly Solvable Schr"odinger Bridges with Quadratic State Cost](https://mlsanity.com/paper/2407.15245/) (2024)
- [Isomorphism Properties of Optimality and Equilibrium Solutions under Equivalent Information Structure Transformations I: Stochastic Dynamic Teams](https://mlsanity.com/paper/2104.05787/) (2021)

## Built on by (most cited first, 11 of 11)

- [NETS: A Non-Equilibrium Transport Sampler](https://mlsanity.com/paper/2410.02711/) (2024)
- [Non-equilibrium Annealed Adjoint Sampler](https://mlsanity.com/paper/2506.18165/) (2025)
- [On the Hopf-Cole Transform for Control-affine Schr"odinger Bridge](https://mlsanity.com/paper/2503.17640/) (2025)
- [Generative Stochastic Optimal Transport: Guided Harmonic Path-Integral Diffusion](https://mlsanity.com/paper/2512.11859/) (2025)
- [Steering Dynamical Regimes of Diffusion Models by Breaking Detailed Balance](https://mlsanity.com/paper/2602.15914/) (2026)
- [Sampling Decisions: Exact Path-Space Control for Physics-Informed Generative Sampling](https://mlsanity.com/paper/2503.14549/) (2025)
- [Mean-Field Path-Integral Diffusion: From Samples to Interacting Agents](https://mlsanity.com/paper/2605.00007/) (2026)
- [Adaptive Path Integral Diffusion: AdaPID](https://mlsanity.com/paper/2512.11858/) (2025)
- [Analytic Bridge Diffusions for Controlled Path Generation](https://mlsanity.com/paper/2605.02961/) (2026)
- [Temporal Memory for Resource-Constrained Agents: Continual Learning via Stochastic Compress-Add-Smooth](https://mlsanity.com/paper/2604.00067/) (2026)
- [iADD: Improving Alignment and Diversity in Diffusion Policy Optimization](https://mlsanity.com/paper/2610.01789/) (2026)

## Abstract (arXiv)

In this manuscript, we present a novel approach for sampling from a continuous multivariate probability distribution, which may either be explicitly known (up to a normalization factor) or represented via empirical samples. Our method constructs a time-dependent bridge from a delta function centered at the origin of the state space at t=0, optimally transforming it into the target distribution at t=1. We formulate this as a Stochastic Optimal Control problem of the Path Integral Control type, with a cost function comprising (in its basic form) a quadratic control term, a quadratic state term, and a terminal constraint. This framework, which we refer to as Harmonic Path Integral Diffusion (H-PID), leverages an analytical solution through a mapping to an auxiliary quantum harmonic oscillator in imaginary time. The H-PID framework results in a set of efficient sampling algorithms, without the incorporation of Neural Networks. The algorithms are validated on two standard use cases: a mixture of Gaussians over a grid and images from CIFAR-10. The transparency of the method allows us to analyze the algorithms in detail, particularly revealing that the current weighted state is an order parameter for the dynamic phase transition, signaling earlier, at t<1, that the sample generation process is almost complete. We contrast these algorithms with other sampling methods, particularly simulated annealing and path integral sampling, highlighting their advantages in terms of analytical control, accuracy, and computational efficiency on benchmark problems. Additionally, we extend the methodology to more general cases where the underlying stochastic differential equation includes an external deterministic, possibly non-conservative force, and where the cost function incorporates a gauge potential term.