A Heterogeneous High Dimensional Approximate Nearest Neighbor Algorithm

Abstract

We consider the problem of finding high dimensional approximate nearest neighbors. Suppose there are d independent rare features, each having its own independent statistics. A point x will have x_{i}=0 denote the absence of feature i, and x_{i}=1 its existence. Sparsity means that usually x_{i}=0. Distance between points is a variant of the Hamming distance. Dimensional reduction converts the sparse heterogeneous problem into a lower dimensional full homogeneous problem. However we will see that the converted problem can be much harder to solve than the original problem. Instead we suggest a direct approach. It consists of T tries. In try t we rearrange the coordinates in decreasing order of (1-r_{t,i})\frac{p_{i,11}}{p_{i,01}+p_{i,10}} \ln\frac{1}{p_{i,1*}} where 0<r_{t,i}<1 are uniform pseudo-random numbers, and the p's are the coordinate's statistical parameters. The points are lexicographically ordered, and each is compared to its neighbors in that order. We analyze a generalization of this algorithm, show that it is optimal in some class of algorithms, and estimate the necessary number of tries to success. It is governed by an information like function, which we call bucketing forest information. Any doubts whether it is "information" are dispelled by another paper, where unrestricted bucketing information is defined.