Attention in Geometry: Scalable Spatial Modeling via Adaptive Density Fields and FAISS-Accelerated Kernels

Abstract

Spatial computation in geographic systems increasingly requires query-conditioned, local, interpretable aggregation under metric constraints. Many classical approaches rely on global summation and treat approximation as an implementation concern, limiting interpretability and scalability at large scales. We propose the Adaptive Density Field (ADF), a geometric attention framework that formulates spatial aggregation as a query-conditioned, metric-induced attention operator in continuous space. Given a set of labelled spatial points with associated scalar scores, ADF defines a continuous intensity field over space. For a given query location, the field value is obtained via a local adaptive Gaussian kernel mixture centered on the query's nearest neighbors, where kernel bandwidths are modulated by point-specific scores to evaluate local aggregated influence. Additionally, approximate nearest-neighbor search is introduced, enabling scalable execution while preserving locality. The proposed ADF bridges concepts from adaptive kernel methods, classical GIS methods, and attention mechanisms by reinterpreting spatial influence as geometry-embedded attention, grounded in physical distance rather than learned latent projections. The proposed framework is formulation-level rather than algorithm-specific, allowing flexible kernel choices, score-to-bandwidth mappings, and approximation parameters. This approach provides a unifying perspective on spatial influence modeling that emphasizes structure, scalability, and geometric interpretability, with relevance to geographic information systems and spatial machine learning.