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NeuroHex: Highly-Efficient Hex Coordinate System for Creating World Models to Enable Adaptive AI

Quinn Jacobson, Joe Luo, Jingfei Xu, Shanmuga Venkatachalam, Kevin Wang, Dingchao Rong, John Paul Shen

TL;DR

Initial results, based on actual city and neighborhood scale data sets, demonstrate that NeuroHex offers a highly efficient substrate for building dynamic world models to enable adaptive spatial reasoning in autonomous AI systems with continuous online learning capability.

Abstract

NeuroHex is a hexagonal coordinate system designed to support highly efficient world models and reference frames for online adaptive AI systems. Inspired by the hexadirectional firing structure of grid cells in the human brain, NeuroHex adopts a cubic isometric hexagonal coordinate formulation that provides full 60° rotational symmetry and low-cost translation, rotation and distance computation. We develop a mathematical framework that incorporates ring indexing, quantized angular encoding, and a hierarchical library of foundational, simple, and complex geometric shape primitives. These constructs allow low-overhead point-in-shape tests and spatial matching operations that are expensive in Cartesian coordinate systems. To support realistic settings, the NeuroHex framework can process OpenStreetMap (OSM) data sets using an OSM-to-NeuroHex (OSM2Hex) conversion tool. The OSM2Hex spatial abstraction processing pipeline can achieve a reduction of 90-99% in geometric complexity while maintaining the relevant spatial structure map for navigation. Our initial results, based on actual city and neighborhood scale data sets, demonstrate that NeuroHex offers a highly efficient substrate for building dynamic world models to enable adaptive spatial reasoning in autonomous AI systems with continuous online learning capability.

NeuroHex: Highly-Efficient Hex Coordinate System for Creating World Models to Enable Adaptive AI

TL;DR

Initial results, based on actual city and neighborhood scale data sets, demonstrate that NeuroHex offers a highly efficient substrate for building dynamic world models to enable adaptive spatial reasoning in autonomous AI systems with continuous online learning capability.

Abstract

NeuroHex is a hexagonal coordinate system designed to support highly efficient world models and reference frames for online adaptive AI systems. Inspired by the hexadirectional firing structure of grid cells in the human brain, NeuroHex adopts a cubic isometric hexagonal coordinate formulation that provides full 60° rotational symmetry and low-cost translation, rotation and distance computation. We develop a mathematical framework that incorporates ring indexing, quantized angular encoding, and a hierarchical library of foundational, simple, and complex geometric shape primitives. These constructs allow low-overhead point-in-shape tests and spatial matching operations that are expensive in Cartesian coordinate systems. To support realistic settings, the NeuroHex framework can process OpenStreetMap (OSM) data sets using an OSM-to-NeuroHex (OSM2Hex) conversion tool. The OSM2Hex spatial abstraction processing pipeline can achieve a reduction of 90-99% in geometric complexity while maintaining the relevant spatial structure map for navigation. Our initial results, based on actual city and neighborhood scale data sets, demonstrate that NeuroHex offers a highly efficient substrate for building dynamic world models to enable adaptive spatial reasoning in autonomous AI systems with continuous online learning capability.
Paper Structure (24 sections, 9 figures, 1 table)

This paper contains 24 sections, 9 figures, 1 table.

Table of Contents

  1. Introduction
  2. Background
  3. Cortical Columns and Reference Frames
  4. Hexagonal Coordinate Systems
  5. Neuroscience Background and Inspiration
  6. Hexagonal Coordinate Systems
  7. Working Model of The Brain and Motivation for NeuroHex
  8. NeuroHex: Computationally Efficient Hexagonal Coordinates Systems
  9. Mathematical Foundations of the Hexagonal Space
  10. NeuroHex Ring Encoding, Positional, Angular Arithmetics
  11. NeuroHex Shape Library
  12. Foundational Shapes
  13. Simple Shapes
  14. Complex Shapes
  15. NeuroHex Translation, Rotation, and Scaling of Shapes
  16. ...and 9 more sections

Figures (9)

  • Figure 1: Hexagonal Coordinate Systems: H3Geo (left) with positive coordinates (i,j,k) H3Geo; our NeuroHex (right) with positive and negative coordinates (q,r,s).
  • Figure 2: High-Level working model of the brain with both Egocentric and Allocentric perspectives involving Grid Cells, adaptive World Models, and hierarchy of Reference Frames.
  • Figure 3: Hierarchical Reference Frames used for building maps of physical world based on diverse sensory modalities.
  • Figure 4: Foundational Components in the NeuroHex coordinate system: Wedge, Ring, and Point.
  • Figure 5: Distance Computation (left) and Angular Direction Quantization (right) in the NeuroHex coordinate system.
  • ...and 4 more figures