Scale-Invariant Ergodic MMD (Scale-EMMD)

"Search at Scale: Improving Numerical Conditioning of Ergodic Coverage Optimization for Multi-Scale Domains"

Code repository for the paper Search at Scale: Improving Numerical Conditioning of Ergodic Coverage Optimization for Multi-Scale Domains.

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Overview

This repository implements trajectory optimization algorithms for scale-invariant ergodic coverage on 3D domains. Trajectories are optimized to match a target information distribution (defined over a point cloud or mesh) while respecting a velocity constraint, using Maximum Mean Discrepancy (MMD) as the coverage objective.

Three techniques are provided to improve numerical conditioning across multi-scale domains:

Method	Script	Key Feature
SI-EMMD	scale_invariance/example.py	Normalizes domain extent prior to optimization
Annealing	annealing/example.py	Cosine-anneals bandwidth from coarse to fine resolution
LSE-MMD	lse/example.py	Replaces sums with log-sum-exp for numerical stability

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Repository Structure

scale_emmd/
├── methods/
│   ├── emmd_lib.py          # Core Scale_EMMD solver class
│   └── solver_anneal.py     # Augmented Lagrangian optimizer
├── annealing/
│   └── example.py           # Bandwidth annealing demo
├── lse/
│   └── example.py           # LSE loss vs. standard MMD demo
├── scale_invariance/
│   └── example.py           # Scale-invariant solver demo
├── obj_files/
│   └── bunny_mesh.obj       # Stanford Bunny mesh (test geometry)
├── LICENSE
└── README.md

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Installation

Requirements: Python 3.9+

pip install jax jaxopt numpy matplotlib open3d trimesh

GPU acceleration: For large trajectories or meshes, JAX with GPU support is recommended. See the JAX installation guide for CUDA setup.

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Quick Start

All examples run from the repository root. Each script contains a clearly marked configuration block at the top of __main__ — edit those variables to change behavior.

1. Scale-Invariant EMMD

Compares standard EMMD against the scale-invariant variant on a scaled 3D mesh.

python scale_invariance/example.py

Configuration block (scale_invariance/example.py):

method = 'si-emmd'  # Options: 'emmd', 'si-emmd'
scale = 5.0         # mesh scale
h = 0.05            # kernel bandwidth
T = 800             # trajectory length

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2. Bandwidth Annealing

Solves coverage on a 3D mesh by annealing the bandwidth from a broad initial value down to one matched to the sensor radius, using a cosine schedule.

python annealing/example.py

Configuration block (annealing/example.py):

method = 'anneal'   # Options: 'emmd', 'anneal'
h = 0.05            # initial bandwidth
sensor_rad = 1.5    # target sensing radius
scale = 10.0        # mesh scale
T = 1000            # trajectory length

The target bandwidth is derived automatically as h_target = sensor_rad² / |log(ε)| with ε = 0.05.

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3. LSE Loss Variant

Runs the annealing solver with a log-sum-exp (numerically stable) loss in place of the standard MMD sum.

python lse/example.py

Configuration block (lse/example.py):

lse = True          # True: log-sum-exp loss;  False: standard MMD
scale = 100.0       # mesh scale
h = 0.05            # starting bandwidth for annealing
sensor_rad = 1.5    # target sensing radius
T = 1000            # trajectory length

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Citation

If you use this code in your research, please cite:

@article{scale_emmd_2026,
  title   = {Search at Scale: Improving Numerical Conditioning of Ergodic
             Coverage Optimization for Multi-Scale Domains},
  author  = {},
  year    = {2026},
  note    = {Intelligent Autonomy Robotics Lab}
}

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License

MIT License — Copyright (c) 2026 Intelligent Autonomy Robotics Lab. See LICENSE for details.