Scaling Laws for Grid-Based Approximate Nearest Neighbor Search in High Dimensions

Grid-based approaches to approximate nearest neighbor (ANN) search have been absent from modern scaling analyses. We present a systematic characterization of a multiprobe grid algorithm with respect to dataset size $N$ and dimensionality $d$. Our experiments reveal a previously unreported $d$-scaling crossover on the GloVe embedding family, in which multiprobe grid search maintains an approximately constant dimensional scaling exponent while other graph-, tree-, and partitioning-based methods exhibit degrading throughput. The advantage comes with near-linear query scaling in $N$, but also with lower indexing cost than competing ANN methods. Our results suggest that grid-based methods such as multiprobe grid may be competitive in rebuild-heavy or high-dimensional settings where indexing cost and dimensional robustness dictate performance. More broadly, recent work has formalized self-attention as an ANN operation. Thus, the $N$- and $d$-scaling properties of ANN algorithms may guide cost analysis of efficient transformer architectures. Code is available at: https://github.com/weiz345/MultiProbeANN.

Aperon Technical Report: Hierarchical No-Pointer Tangent-Local Search for High-Dimensional Approximate Nearest Neighbors

We present HNTL (Hierarchical No-pointer Tangent-Local), the core vector indexing and candidate generation framework of the Aperon vector memory system. Proximity graphs (e.g., HNSW) incur a heavy pointer tax in memory overhead and induce irregular memory accesses that stall CPU pipelines. HNTL resolves this by partitioning the high-dimensional space into local, coherent grains, representing vectors as low-dimensional coordinates on local tangent spaces, and scanning them sequentially using a pointerless Block-SoA (Structure-of-Arrays) layout. On anisotropic manifold data (d=768, N=10,000), local PCA captures 96.3% of the variance, allowing HNTL to achieve a final Rerank Recall@10 of 1.0000 with a candidate pool size of only C=20 vectors. Hardware profiling via Apple kperf CPU Performance Monitoring Unit (PMU) counters demonstrates a 3.61x speedup (4.137 ns/vector vs. 14.951 ns/vector) for our NEON auto-vectorized C++ Block-SoA scan engine over standard pointer-chasing graph traversals, driven by a 3.59x IPC (Instructions Per Cycle) and near-zero L1/L2 data cache misses.