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

• URL: http://arxiv.org/abs/2601.06135v1
• Date: Mon, 05 Jan 2026 05:42:40 GMT
• Title: Attention in Geometry: Scalable Spatial Modeling via Adaptive Density Fields and FAISS-Accelerated Kernels
• Abstract summary: 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.<n>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.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• 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.

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