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

TL;DR

This paper presents Adaptive Density Fields (ADF), a scalable geometric attention framework that models spatial relationships using continuous, query-conditioned attention, enhanced by FAISS-accelerated nearest-neighbor search, demonstrated on aircraft trajectory analysis.

Contribution

Introduces ADF, a novel geometric attention method that integrates adaptive kernel concepts with scalable FAISS-based indexing for continuous spatial modeling.

Findings

Revealed recurrent airspace structures in aircraft trajectories.

Demonstrated scalable spatial attention with FAISS acceleration.

Extracted meaningful Zones of Influence in trajectory data.

Abstract

This work introduces Adaptive Density Fields (ADF), a geometric attention framework that formulates spatial aggregation as a query-conditioned, metric-induced attention operator in continuous space. By reinterpreting spatial influence as geometry-preserving attention grounded in physical distance, ADF bridges concepts from adaptive kernel methods and attention mechanisms. Scalability is achieved via FAISS-accelerated inverted file indices, treating approximate nearest-neighbor search as an intrinsic component of the attention mechanism. We demonstrate the framework through a case study on aircraft trajectory analysis in the Chengdu region, extracting trajectory-conditioned Zones of Influence (ZOI) to reveal recurrent airspace structures and localized deviations.