# Sampling in Constrained Domains with Orthogonal-Space Variational Gradient Descent

arXiv: https://arxiv.org/abs/2210.06447 · HTML: https://mlsanity.com/paper/2210.06447/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- Builds on 13 of its 42 references that are in the map.
- First posted on arXiv 12 Oct 2022.

## Builds on (most cited first, 13 of 13)

- [Generative Modeling by Estimating Gradients of the Data Distribution](https://mlsanity.com/paper/1907.05600/) (2019)
- [Stein Variational Gradient Descent: A General Purpose Bayesian Inference Algorithm](https://mlsanity.com/paper/1608.04471/) (2016)
- [Cyclical Stochastic Gradient MCMC for Bayesian Deep Learning](https://mlsanity.com/paper/1902.03932/) (2019)
- [Minimax Pareto Fairness: A Multi Objective Perspective](https://mlsanity.com/paper/2011.01821/) (2020)
- [Stein Variational Gradient Descent as Gradient Flow](https://mlsanity.com/paper/1704.07520/) (2017)
- [Bayesian Neural Network Priors Revisited](https://mlsanity.com/paper/2102.06571/) (2021)
- [Measuring Sample Quality with Kernels](https://mlsanity.com/paper/1703.01717/) (2017)
- [On the geometry of Stein variational gradient descent](https://mlsanity.com/paper/1912.00894/) (2019)
- [A Spectral Energy Distance for Parallel Speech Synthesis](https://mlsanity.com/paper/2008.01160/) (2020)
- [Accuracy and Fairness Trade-offs in Machine Learning: A Stochastic Multi-Objective Approach](https://mlsanity.com/paper/2008.01132/) (2020)
- [Convergence Rate of Riemannian Hamiltonian Monte Carlo and Faster Polytope Volume Computation](https://mlsanity.com/paper/1710.06261/) (2017)
- [Sampling with Riemannian Hamiltonian Monte Carlo in a Constrained Space](https://mlsanity.com/paper/2202.01908/) (2022)
- [Fast Convergence of Langevin Dynamics on Manifold: Geodesics meet Log-Sobolev](https://mlsanity.com/paper/2010.05263/) (2020)

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

- [Constrained Stein Variational Trajectory Optimization](https://mlsanity.com/paper/2308.12110/) (2023)
- [Constrained Exploration via Reflected Replica Exchange Stochastic Gradient Langevin Dynamics](https://mlsanity.com/paper/2405.07839/) (2024)
- [Learning Rate Free Sampling in Constrained Domains](https://mlsanity.com/paper/2305.14943/) (2023)
- [A Note on Bias to Complete](https://mlsanity.com/paper/2402.11710/) (2024)
- [Wasserstein gradient flow for optimal probability measure decomposition](https://mlsanity.com/paper/2406.00914/) (2024)
- [Manifold Sampling via Entropy Maximization](https://mlsanity.com/paper/2605.12338/) (2026)
- [Posterior sampling with CNN-based, Plug-and-Play regularization with applications to Post-Stack Seismic Inversion](https://mlsanity.com/paper/2212.14595/) (2022)
- [Functional Gradient Flows for Constrained Sampling](https://mlsanity.com/paper/2410.23170/) (2024)
- [A Taxonomy of Self-Handover](https://mlsanity.com/paper/2504.04939/) (2025)
- [Constrained Stein Variational Gradient Descent for Robot Perception, Planning, and Identification](https://mlsanity.com/paper/2506.00589/) (2025)
- [Efficient Diffusion Models under Nonconvex Equality and Inequality constraints via Landing](https://mlsanity.com/paper/2604.17838/) (2026)

## Abstract (arXiv)

Sampling methods, as important inference and learning techniques, are typically designed for unconstrained domains. However, constraints are ubiquitous in machine learning problems, such as those on safety, fairness, robustness, and many other properties that must be satisfied to apply sampling results in real-life applications. Enforcing these constraints often leads to implicitly-defined manifolds, making efficient sampling with constraints very challenging. In this paper, we propose a new variational framework with a designed orthogonal-space gradient flow (O-Gradient) for sampling on a manifold G0 defined by general equality constraints. O-Gradient decomposes the gradient into two parts: one decreases the distance to G0 and the other decreases the KL divergence in the orthogonal space. While most existing manifold sampling methods require initialization on G0, O-Gradient does not require such prior knowledge. We prove that O-Gradient converges to the target constrained distribution with rate widetildeO(1/the number of iterations) under mild conditions. Our proof relies on a new Stein characterization of conditional measure which could be of independent interest. We implement O-Gradient through both Langevin dynamics and Stein variational gradient descent and demonstrate its effectiveness in various experiments, including Bayesian deep neural networks.