Self-Balancing, Memory Efficient, Dynamic Metric Space Data Maintenance, for Rapid Multi-Kernel Estimation

TL;DR

This work introduces a self-balancing, memory-efficient dynamic octree with a two-parameter $(K,\alpha)$ balance that guarantees logarithmic-time updates and queries in evolving metric spaces. By dynamically adapting partitions and maintaining neighborhood relationships, the approach delivers exponential speedups across SVGD, incremental KNN, RAG, and OT-Flow applications while preserving accuracy. The dynamic octree enables scalable, streaming-friendly handling of high-dimensional data as distributions shift during training and inference. Overall, this framework provides a unified computational backbone for efficient, structure-preserving navigation of generative spaces in modern machine learning pipelines.

Abstract

We present a dynamic self-balancing octree data structure that enables efficient neighborhood maintenance in evolving metric spaces, a key challenge in modern machine learning systems. Many learning and generative models operate as dynamical systems whose representations evolve during training, requiring fast, adaptive spatial organization. Our two-parameter octree supports logarithmic-time updates and queries, eliminating the need for costly full rebuilds as data distributions shift. We demonstrate its effectiveness in four areas: (1) accelerating Stein variational gradient descent by supporting more particles with lower overhead; (2) enabling real-time, incremental KNN classification with logarithmic complexity; (3) facilitating efficient, dynamic indexing and retrieval for retrieval-augmented generation; and (4) improving sample efficiency by jointly optimizing input and latent spaces. Across all applications, our approach yields exponential speedups while preserving accuracy, particularly in high-dimensional spaces where maintaining adaptive spatial structure is critical.

Figures (31)

• Figure 1: Illustration of the $(K,\alpha)$ dynamic octree's adaptive refinement. The spatial view (left) shows finer subdivisions (darker blue) in high-density areas, while the tree structure (right) reveals deeper branches only in dense regions. This reveals how the octree optimizes itself to use the minimum possible depth—creating the most efficient representation with the fewest nodes. By maintaining the minimum necessary internal and leaf nodes, our structure achieves logarithmic-time operations despite non-uniform distributions, delivering optimal memory usage and computational efficiency for evolving metric spaces.
• Figure 2: Adaptive spatial partitioning in incremental KNN classification. Panels show classifier evolution as new data batches are incorporated, with octree structure adapting to data density and class boundaries, maintaining high accuracy and efficiency.
• Figure 3: Build time comparison showing our octree's near-linear scaling.
• Figure 4: Neighborhood list construction with up to 14.3× performance advantage at scale.
• Figure 5: Update time showing our approach's consistent efficiency regardless of data size.