Lightweight-yet-efficient Revitalizing Ball-tree For Point-to-hyperplane Nearest Neighbor Search

Huang Qiang, Tung Anthony K. H.. Arxiv 2023

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Finding the nearest neighbor to a hyperplane (or Point-to-Hyperplane Nearest Neighbor Search, simply P2HNNS) is a new and challenging problem with applications in many research domains. While existing state-of-the-art hashing schemes (e.g., NH and FH) are able to achieve sublinear time complexity without the assumption of the data being in a unit hypersphere, they require an asymmetric transformation, which increases the data dimension from \(d\) to \(Ω(d^2)\). This leads to considerable overhead for indexing and incurs significant distortion errors. In this paper, we investigate a tree-based approach for solving P2HNNS using the classical Ball-Tree index. Compared to hashing-based methods, tree-based methods usually require roughly linear costs for construction, and they provide different kinds of approximations with excellent flexibility. A simple branch-and-bound algorithm with a novel lower bound is first developed on Ball-Tree for performing P2HNNS. Then, a new tree structure named BC-Tree, which maintains the Ball and Cone structures in the leaf nodes of Ball-Tree, is described together with two effective strategies, i.e., point-level pruning and collaborative inner product computing. BC-Tree inherits both the low construction cost and lightweight property of Ball-Tree while providing a similar or more efficient search. Experimental results over 16 real-world data sets show that Ball-Tree and BC-Tree are around 1.1\(\sim\)10\(\times\) faster than NH and FH, and they can reduce the index size and indexing time by about 1\(\sim\)3 orders of magnitudes on average. The code is available at \url{https://github.com/HuangQiang/BC-Tree}.

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