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Geographic Location Encoding with Spherical Harmonics and Sinusoidal Representation Networks
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Proposes spherical harmonic positional embeddings combined with SirenNet for global location encoding, showing SirenNets act as learned Double Fourier Sphere embeddings and reporting state-of-the-art results on checkerboard, land-ocean, ERA5, and iNaturalist 2018 tasks.
Problem
Location encoders for globally distributed geographic data often assume a rectangular lon/lat domain, causing artifacts near the poles; the choice of neural network paired with positional embeddings is also under-explored.
Approach
Use real spherical harmonic basis functions (eigenfunctions of the sphere Laplacian) as well-defined global positional embeddings, paired with sinusoidal representation networks (SirenNet), and show that a SirenNet with direct coordinates can be written as a learned DFS-style embedding.
Key results
- On the checkerboard task, SH with SirenNet reaches 95.8 ± 0.0% accuracy, the highest among tested combinations.
- On ERA5 interpolation, SH with FcNet attains MSE 0.58 ± 0.02 versus 1.38 ± 0.03 for the next best (SphereC+ with FcNet).
- On iNaturalist 2018, SH with SirenNet gives +12.3 ± 0.2% top-1 gain over the image-only 59.2% baseline, slightly above other encoders within standard deviations.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Replace the strongest non-SH embedding with SH | Linear location encoder · Checkerboard classification | accuracy: 64.0 \pm 0.1 -> 94.8 \pm 0.1 | Helped |
| Replace Grid with SH | FcNet location encoder · Checkerboard classification | accuracy: 93.5 \pm 0.3 -> 94.9 \pm 0.1 | Helped |
| Replace Cartesian3D with SH | SirenNet location encoder · Checkerboard classification | accuracy: 95.7 \pm 0.2 -> 95.8 \pm 0.0 | No clear effect |
| Replace Theory with SH | Linear location encoder · Land-Ocean classification | accuracy: 86.9 \pm 0.1 -> 94.4 \pm 0.1 | Helped |
| Replace SphereC+ with SH | FcNet location encoder · Land-Ocean classification | accuracy: 95.3 \pm 0.1 -> 95.9 \pm 0.1 | Helped |
| Replace Grid with SH | SirenNet location encoder · Land-Ocean classification | accuracy: 95.5 \pm 0.2 -> 95.8 \pm 0.1 | Helped |
| Replace SphereC+ with SH | Linear location encoder · ERA5 interpolation | MSE: 8.50 \pm 0.02 -> 1.39 \pm 0.02 | Helped |
| Replace SphereC+ with SH | FcNet location encoder · ERA5 interpolation | MSE: 1.38 \pm 0.03 -> 0.58 \pm 0.02 | Helped |
| Replace Cartesian3D with SH | SirenNet location encoder · ERA5 interpolation | MSE: 1.57 \pm 0.11 -> 1.19 \pm 0.04 | Helped |
| Replace SphereC with SH | SirenNet location encoder · iNaturalist 2018 | top-1 accuracy improvement: +12.3 \pm 0.1 -> +12.3 \pm 0.2 | No clear effect |
| Replace SphereC with SH | SirenNet location encoder · iNaturalist 2018 | top-3 accuracy improvement: +8.8 \pm 0.1 -> +9.0 \pm 0.1 | Helped |
| Increase Legendre polynomials from 10 to 20 (ablation) | SH + Linear · Checkerboard resolution | resolvable resolution: 9- and 18-degree resolutions | Helped |
Surprising findings
- A SirenNet with direct (identity) lon/lat input performs competitively, e.g., 94.7% on checkerboard, which the authors explain as a learned DFS embedding.
- Spherical harmonics with a simple linear layer alone performs reasonably well across datasets, though SphereM and Theory variants vary strongly by network.
Limitations
- Iterative closed-form spherical harmonic computation scales poorly with large L, requiring the analytic implementation.
- FcNet and SirenNet rankings vary by dataset; no single neural network wins everywhere.
- On iNaturalist, differences between top encoders are often within standard deviations.
Experimental setup
Datasets: Fibonacci-lattice checkerboard (synthetic) Natural Earth land shapefile (land-ocean) ERA5 reanalysis (Jan 1 2018) iNaturalist 2018
Benchmarks: Fibonacci-lattice checkerboard Land-Ocean classification ERA5 interpolation iNaturalist 2018
Metrics: classification accuracy mean squared error relative top-1 and top-3 accuracy increase inference runtime
Baselines: Direct Cartesian3D Wrap Grid Theory SphereC SphereC+ SphereM SphereM+ FcNet Linear
Builds on: Double Fourier Sphere SirenNet Wrap positional encoding Grid location encoder SphereM+
Closest papers by meaning
- Poly2Vec: Polymorphic Fourier-Based Encoding of Geospatial Objects for GeoAI Applications (ICML 2025 · similarity 0.86)
- TorchSpatial: A Location Encoding Framework and Benchmark for Spatial Representation Learning (NeurIPS 2024 · similarity 0.85)
- LocDiff: Identifying Locations on Earth by Diffusing in the Hilbert Space (NeurIPS 2025 · similarity 0.85)
- SatCLIP: Global, General-Purpose Location Embeddings with Satellite Imagery (AAAI 2025 · similarity 0.85)
- Hybrid Neural Representations for Spherical Data (ICML 2024 · similarity 0.85)
- No Location Left Behind: Measuring and Improving the Fairness of Implicit Representations for Earth Data (ICLR 2025 · similarity 0.85)
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