Learnable method detects spatial patterns in complex graph data
SAIL: A Learnable Generalisation of Spatial Autocorrelation over Graph-Structured Data
Computer Vision and Pattern Recognition
Summary
When studying data spread over space, it’s important to know if values are random or form patterns. Existing tools for measuring spatial patterns only work on simple data types and fixed connections. The authors created SAIL, a flexible and trainable method that learns how features and connections create spatial patterns, even for complex, high-dimensional data. SAIL can explain local and global spatial patterns, predict outcomes where space matters, and find interesting hotspots in data like images or biological networks.
What this means in practice
- •For biomedical data scientists: Detect coherent spatial patterns and hotspots in large biomedical interaction networks to improve biological insights.
- •For geospatial analysts: Identify meaningful spatial structures in complex urban or environmental graphs with learnable neighbourhood relationships.
Authors
Piotr Keller, Fayyaz Minhas
Abstract
In vision and scientific domains involving spatial data, a key question is whether node feature values vary independently across space or form coherent spatial structures. Classical spatial autocorrelation statistics such as Moran's $I$ and Geary's $C$ quantify this phenomenon but operate on a single scalar variable with fixed neighbourhood weights, limiting their use for high-dimensional graph-like data. We introduce SAIL (Spatial Autocorrelation Index Learner), a differentiable framework that generalises spatial autocorrelation into a learnable operator by jointly learning feature projections defining candidate scalar fields and inter-node neighbourhood affinities. SAIL produces interpretable local and global spatial coherence descriptors, enabling supervised prediction and unsupervised discovery of spatial autocorrelation modes and hotspots. We show that SAIL exactly recovers classical spatial statistics under appropriate parametrisations and, for tasks driven by latent spatial autocorrelation, can approximate the Bayes-optimal predictor with an inductive bias matched to such tasks. Experiments on synthetic data and large-scale biomedical graphs demonstrate statistical correctness and discovery of spatial coherence with applications across vision, geospatial analysis, and biomedicine. The code repository is available at: https://github.com/pkeller00/SAIL.