Localized, High-resolution Geographic Representations with Slepian Functions

TL;DR

The paper tackles the challenge that geographic data are inherently local and global encoders struggle to capture fine-grained regional patterns. It introduces Slepian-based regionally concentrated encoders and a hybrid Slepian-SH architecture to blend high-resolution local detail with global context, enabling scalable, pole-safe representations on the sphere. Through extensive experiments across regression, classification, and geo-aware image tasks, the authors show that Slepian encodings yield higher predictive performance with far fewer embedding dimensions than global SH bases, and that the hybrid approach consistently outperforms either component alone. They also extend the framework to spatio-temporal data using DPSS (Discret Prolate Spheroidal Sequences) to capture temporal bandwidth without leakage. Overall, the work demonstrates a robust, efficient strategy for high-resolution, region-specific geographic representations with broad practical impact in geospatial ML.

Abstract

Geographic data is fundamentally local. Disease outbreaks cluster in population centers, ecological patterns emerge along coastlines, and economic activity concentrates within country borders. Machine learning models that encode geographic location, however, distribute representational capacity uniformly across the globe, struggling at the fine-grained resolutions that localized applications require. We propose a geographic location encoder built from spherical Slepian functions that concentrate representational capacity inside a region-of-interest and scale to high resolutions without extensive computational demands. For settings requiring global context, we present a hybrid Slepian-Spherical Harmonic encoder that efficiently bridges the tradeoff between local-global performance, while retaining desirable properties such as pole-safety and spherical-surface-distance preservation. Across five tasks spanning classification, regression, and image-augmented prediction, Slepian encodings outperform baselines and retain performance advantages across a wide range of neural network architectures.

Figures (13)

• Figure 1: Slepian functions concentrate a band-limited basis inside a chosen region. Traditional geographic representations (left) distribute a fixed resolution budget uniformly, forcing a trade-off between global smoothness and local detail. Our proposed hybrid Slepian encoder (right) concentrates high-frequency basis functions exclusively within a region-of-interest (red circle) while preserving global context outside.
• Figure 2: Constructing our hybrid Slepian encoder. ( ) For a region encompassing the geographic coordinate, we compute Slepian eigenfunctions $\{g_n\}$ ordered by their eigenvalues $\mu_n$ and discard modes with eigenvalues under the regional Shannon number to form our Slepian positional encoder. These modes are concatenated with global SH basis functions ( ) of a coarse resolution to form our hybrid positional embedding ( ). The hybrid positional embedding is passed to a neural network to form our final location encoder that captures high-resolution detail within the region-of-interest (ROI) while retaining global context ( ).
• Figure 3: Slepian functions concentrate representational capacity within the cap region. As cap coverage increases, $R^2$ improves monotonically, while global spherical harmonics (dotted line) remain constant regardless of the spatial extent of interest. Error bars show $1 \times$ SD over 5 random seeds.
• Figure 4: Slepian-based location encoders better preserve fine-scale spatial/geographic detail. We visualize Arctic Mean Sea Surface (MSS) interpolation results proposed in chen2024deep. Spherical Harmonics at $L \geq 40$ diverges and does not produce a valid reconstruction. Interpolation results reinforce that Slepians inherit the pole-safe property of SH (\ref{['sec:pole_safety']}) unlike Space2Vec and several DFS-derived location encoders. Line-like artifacts are visible around the poles due to the nature of satellite track measurements. We visualize the interpolation results of additional baselines in \ref{['fig:mss-globes-appendix']}.
• Figure 4: Land-Ocean classification results with our Hybrid Slepian. F1 score for spherical-harmonic (SH) and Hybrid Slepian encoders at two settings of the global bandlimit $L_g$, evaluated on three test-set regimes.