# Fast Ergodic Search with Kernel Functions

arXiv: https://arxiv.org/abs/2403.01536 · HTML: https://mlsanity.com/paper/2403.01536/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 25 of its 69 references that are in the map.
- First posted on arXiv 3 Mar 2024.

## Builds on (most cited first, 25 of 25)

- [A micro Lie theory for state estimation in robotics](https://mlsanity.com/paper/1812.01537/) (2018)
- [You Only Demonstrate Once: Category-Level Manipulation from Single Visual Demonstration](https://mlsanity.com/paper/2201.12716/) (2022)
- [Dynamic Movement Primitives in Robotics: A Tutorial Survey](https://mlsanity.com/paper/2102.03861/) (2021)
- [Active Learning of Dynamics for Data-Driven Control Using Koopman Operators](https://mlsanity.com/paper/1906.05194/) (2019)
- [Robust Multi-Modal Policies for Industrial Assembly via Reinforcement Learning and Demonstrations: A Large-Scale Study](https://mlsanity.com/paper/2103.11512/) (2021)
- [Residual Learning from Demonstration: Adapting DMPs for Contact-rich Manipulation](https://mlsanity.com/paper/2008.07682/) (2020)
- [Contact-Aided Invariant Extended Kalman Filtering for Robot State Estimation](https://mlsanity.com/paper/1904.09251/) (2019)
- [Ergodic Exploration of Distributed Information](https://mlsanity.com/paper/1708.09352/) (2017)
- [Ergodic imitation: Learning from what to do and what not to do](https://mlsanity.com/paper/2103.17098/) (2021)
- [Real-Time Area Coverage and Target Localization using Receding-Horizon Ergodic Exploration](https://mlsanity.com/paper/1708.08416/) (2017)
- [Ergodic Exploration using Tensor Train: Applications in Insertion Tasks](https://mlsanity.com/paper/2101.04428/) (2021)
- [Imitation and Supervised Learning of Compliance for Robotic Assembly](https://mlsanity.com/paper/2111.10488/) (2021)
- [A Note on Monte Carlo Integration in High Dimensions](https://mlsanity.com/paper/2206.09036/) (2022)
- [Decentralized Ergodic Control: Distribution-Driven Sensing and Exploration for Multi-Agent Systems](https://mlsanity.com/paper/1806.05220/) (2018)
- [Prim-LAfD: A Framework to Learn and Adapt Primitive-Based Skills from Demonstrations for Insertion Tasks](https://mlsanity.com/paper/2212.00955/) (2022)
- [Reinforcement learning with Demonstrations from Mismatched Task under Sparse Reward](https://mlsanity.com/paper/2212.01509/) (2022)
- [Time Optimal Ergodic Search](https://mlsanity.com/paper/2305.11643/) (2023)
- [Online Feedback Control for Input-Saturated Robotic Systems on Lie Groups](https://mlsanity.com/paper/1709.00376/) (2017)
- [Invariant Smoothing on Lie Groups](https://mlsanity.com/paper/1803.02076/) (2018)
- [Imitating Human Search Strategies for Assembly](https://mlsanity.com/paper/1809.04860/) (2018)
- [AK: Attentive Kernel for Information Gathering](https://mlsanity.com/paper/2205.06426/) (2022)
- [Scale-Invariant Fast Functional Registration](https://mlsanity.com/paper/2209.12763/) (2022)
- [Safety-Critical Ergodic Exploration in Cluttered Environments via Control Barrier Functions](https://mlsanity.com/paper/2211.04310/) (2022)
- [Characterizing the Uncertainty of Jointly Distributed Poses in the Lie Algebra](https://mlsanity.com/paper/1906.07795/) (2019)
- [An Ergodic Measure for Active Learning From Equilibrium](https://mlsanity.com/paper/2006.03552/) (2020)

## Built on by (most cited first, 5 of 5)

- [Flow Matching Ergodic Coverage](https://mlsanity.com/paper/2504.17872/) (2025)
- [Volumetric Ergodic Control](https://mlsanity.com/paper/2511.11533/) (2025)
- [Belief-Conditioned One-Step Diffusion: Real-Time Trajectory Planning with Just-Enough Sensing](https://mlsanity.com/paper/2508.12166/) (2025)
- [Shortcomings and capacities of real-constrained neural networks in complex spaces](https://mlsanity.com/paper/2606.04390/) (2026)
- [TRACE: Ergodic Trajectory Optimization for Active Scene Reconstruction](https://mlsanity.com/paper/2608.02304/) (2026)

## Abstract (arXiv)

Ergodic search enables optimal exploration of an information distribution while guaranteeing the asymptotic coverage of the search space. However, current methods typically have exponential computation complexity in the search space dimension and are restricted to Euclidean space. We introduce a computationally efficient ergodic search method. Our contributions are two-fold. First, we develop a kernel-based ergodic metric and generalize it from Euclidean space to Lie groups. We formally prove the proposed metric is consistent with the standard ergodic metric while guaranteeing linear complexity in the search space dimension. Secondly, we derive the first-order optimality condition of the kernel ergodic metric for nonlinear systems, which enables efficient trajectory optimization. Comprehensive numerical benchmarks show that the proposed method is at least two orders of magnitude faster than the state-of-the-art algorithm. Finally, we demonstrate the proposed algorithm with a peg-in-hole insertion task. We formulate the problem as a coverage task in the space of SE(3) and use a 30-second-long human demonstration as the prior distribution for ergodic coverage. Ergodicity guarantees the asymptotic solution of the peg-in-hole problem so long as the solution resides within the prior information distribution, which is seen in the 100% success rate.