Self-balancing, Memory Efficient, Dynamic Metric Space Data Maintenance, for Rapid Multi-kernel Estimation

We present a dynamic self-balancing octree data structure that fundamentally transforms neighborhood maintenance in evolving metric spaces. Learning systems, from deep networks to reinforcement learning agents, operate as dynamical systems whose trajectories through high-dimensional spaces require efficient importance sampling for optimal convergence. Generative models operate as dynamical systems whose latent representations cannot be learned in one shot, but rather grow and evolve sequentially during training—requiring continuous adaptation of spatial relationships. Our two-parameter \((K, \alpha )\) dynamic octree addresses this challenge by providing a computational fabric that efficiently organizes both the generation flow and querying flow operating on different time scales by enabling logarithmic-time updates and queries without requiring complete rebuilding as distributions evolve. We demonstrate its effectiveness across four key applications: (1) accelerating Stein Variational Gradient Descent by enabling larger particle sets with reduced computation; (2) supporting real-time incremental KNN classification with logarithmic updates; (3) improving retrieval-augmented generation by enabling efficient, incremental semantic indexing; and (4) showing that maintaining both input and latent space structures accelerates convergence and improves sample efficiency. Across all applications, our experimental results confirm exponential performance improvements over standard methods while maintaining accuracy. These improvements are particularly significant for high-dimensional spaces where efficient neighborhood maintenance is crucial to navigate complex latent manifolds. By providing guaranteed logarithmic bounds for both update and query operations, our approach enables more data-efficient solutions to previously computationally prohibitive problems, establishing a new approach to dynamic spatial relationship maintenance in machine learning.