Localized, High-Resolution Geographic Representations with Slepian Functions

^{Slepian functions concentrate representational capacity where it matters. Left: Global spherical harmonics (L=10) spread energy uniformly. Right: Our hybrid encoder adds Slepian modes concentrated on a region of interest (India), building fine-grained local detail while preserving global context.}

Overview

Geographic data is fundamentally local. Disease outbreaks cluster in population centers, ecological patterns emerge along coastlines, and economic activity concentrates within administrative boundaries. Yet standard location encoders—spherical harmonics, Fourier features, grid embeddings—distribute representational capacity uniformly across the globe. This creates a resolution bottleneck: to capture fine-grained patterns in California, you need high-frequency basis functions everywhere on Earth. Global spherical harmonics become numerically unstable beyond L≈40, and dense global representations scale poorly. Slepian functions solve the concentration problem: finding bandlimited functions that maximize energy inside a region of interest. They provide:

• Spatial concentration — High-resolution detail exactly where you need it
• Compact representations — Dimensionality scales with region size, not the globe
• Numerical stability — Reach L=120+ without floating-point issues
• Pole safety — Well-defined everywhere on the sphere, including polar regions

Our hybrid encoder concatenates:

1. Regional Slepian basis at high bandwidth (L=40–120) for local detail
2. Global spherical harmonics at low bandwidth (L=10) for context

This bridges the local-global tradeoff: concentrated capacity inside the region, smooth global coverage outside.

Structure and Usage

SlepianPosEnc/
├── src/
│   ├── nn/                              # Neural network architectures (MLP, SIREN, ResMLP, GLU)
│   ├── pe_baselines/                    # Baseline positional encoders
│   ├── spherical_harmonics_ylm.py       # Spherical harmonics implementation
│   ├── supervised_geographic_classification/  # Geographic prediction experiments
│   └── slepian_image_augmentation/           # Image-augmented prediction
├── results/                             # Experiment outputs
├── cache/                               # Precomputed Slepian features
└── visualizations/                      # Figures

Experiments

1. Supervised Geographic Prediction

Three tasks comparing Slepian vs baseline positional encoders:

• California Housing - Property value regression
• Japan Prefectures - 47-class geographic classification
• Arctic MSS - Sea surface height regression

See src/supervised_geographic_classification/README.md for details.

2. Image-Augmented Geographic Prediction

Building density regression combining satellite image embeddings with Slepian encodings across 4 global regions.

See src/slepian_image_augmentation/README.md for details.

Requirements

python >= 3.9
torch >= 2.0
numpy, scipy, pandas
matplotlib, scikit-learn
pyshtools              # Slepian function computation

Additional for image augmentation:

earthengine-api        # Google Earth Engine
geopandas, cartopy     # Geospatial visualization

Quick Start

# Clone with submodules
git clone --recursive https://github.com/your-repo/SlepianPosEnc.git
cd SlepianPosEnc

# Run California Housing experiment (no data download needed)
cd src/supervised_geographic_classification/scripts
bash run_california_experiments.sh

Citation

If you find this code or our method useful in your research, please consider citing our paper:

@inproceedings{
rao2026localized,
title={Localized, High-resolution Geographic Representations with Slepian Functions},
author={Anonymous},
booktitle={Forty-third International Conference on Machine Learning},
year={2026},
url={https://openreview.net/forum?id=eWQQ0tO0kB}
}