Which Space Partitioning Tree to Use for Search?

We consider the task of nearest-neighbor search with the class of binary-spacepartitioning trees, which includes kd-trees, principal axis trees and random projection trees, and try to rigorously answer the question “which tree to use for nearestneighbor search?” To this end, we present the theoretical results which imply that trees with better vector quantization performance have better search performance guarantees. We also explore another factor affecting the search performance – margins of the partitions in these trees. We demonstrate, both theoretically and empirically, that large margin partitions can improve tree search performance. 1 Nearest-neighbor search Nearest-neighbor search is ubiquitous in computer science. Several techniques exist for nearestneighbor search, but most algorithms can be categorized into two following groups based on the indexing scheme used – (1) search with hierarchical tree indices, or (2) search with hash-based indices. Although multidimensional binary space-partitioning trees (or BSP-trees), such as kd-trees [1], are widely used for nearest-neighbor search, it is believed that their performances degrade with increasing dimensions. Standard worst-case analyses of search with BSP-trees in high dimensions usually lead to trivial guarantees (such as, an Ω(n) search time guarantee for a single nearest-neighbor query in a set of n points). This is generally attributed to the “curse of dimensionality” – in the worst case, the high dimensionality can force the search algorithm to visit every node in the BSP-tree. However, these BSP-trees are very simple and intuitive, and still used in practice with success. The occasional favorable performances of BSP-trees in high dimensions are attributed to the low “intrinsic” dimensionality of real data. However, no clear relationship between the BSP-tree search performance and the intrinsic data properties is known. We present theoretical results which link the search performance of BSP-trees to properties of the data and the tree. This allows us to identify implicit factors influencing BSP-tree search performance — knowing these driving factors allows us to develop successful heuristics for BSP-trees with improved search performance. Algorithm 1 BSP-tree search Input: BSP-tree T on set S, Query q, Desired depth l Output: Candidate neighbor p current tree depth lc ← 0 current tree node Tc ← T while lc < l do if 〈Tc.w, q〉+ Tc.b ≤ 0 then Tc ← Tc.left child else Tc ← Tc.right child end if Increment depth lc ← lc + 1 end while p← argminr∈Tc∩S ‖q − r‖. Each node in a BSP-tree represents a region of the space and each non-leaf node has a left and right child representing a disjoint partition of this region with some separating hyperplane and threshold (w, b). A search query on this tree is usually answered with a depth-first branch-and-bound algorithm. Algorithm 1 presents a simplified version where a search query is answered with a small set of neighbor candidates of any desired size by performing a greedy depth-first tree traversal to a specified depth. This is known as defeatist tree search. We are not aware of any data-dependent analysis of the quality of the results from defeatist BSP-tree search. However, Verma et al. (2009) [2] presented adaptive data-dependent analyses of some BSP-trees for the task of vector quantization. These results show precise connections between the quantization performance of the BSP-trees and certain properties of the data (we will present these data properties in Section 2).