Barycentric Simplicial Hashing for Approximate Nearest Neighbor Search: A Four-State Topological Hash Competitive with Industry-Standard Product Quantization at Half the Memory

We present Barycentric Simplicial Hashing (BSH), a data-dependent binary indexing method for approximate nearest neighbor (ANN) search that uses the local triangulation of a vector space as a discrete coordinate system. Given a set of database vectors partitioned into Voronoi cells, we construct a k-NN simplicial complex within each cell and assign each vector a compact binary code by evaluating its barycentric zone relative to every triangle in the complex.

The key contribution is a four-state quantization per triangle: a vector is assigned to the zone of the nearest vertex (states 0, 1, 2) or to the barycentre zone (state 3) when the triangle centroid is the closest reference point. Empirical measurement confirms that the barycentre state is activated for over 51% of all triangle-vector assignments in 24-dimensional subspaces, making it the primary discriminator rather than a rare case.

On out-of-sample queries drawn from an independent distribution (separate random seed, never seen during index construction), the four-state hash achieves 84-90% Recall@1 in the top-10% of candidates within a Voronoi cell, using only 34-38 bytes per vector. Industry-standard FAISS IVF-PQ achieves comparable recall (85-90%) at 64 bytes per vector. BSH delivers competitive recall at approximately half the memory footprint. All methods are hardware-agnostic and were empirically validated on ARM Cortex-X3 using NEON intrinsics.