Lightweight-Yet-Efficient: Revitalizing Ball-Tree for Point-to-Hyperplane Nearest Neighbor Search

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 $\Omega(d^2)$. 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}.