Low-Quality Dimension Reduction and High-Dimensional Approximate Nearest Neighbor

The approximate nearest neighbor problem ( -ANN) in Euclidean settings is a fundamental question, which has been addressed by two main approaches: Data-dependent space partitioning techniques perform well when the dimension is relatively low, but are affected by the curse of dimensionality. On the other hand, locality sensitive hashing has polynomial dependence in the dimension, sublinear query time with an exponent inversely proportional to (1 + )2, and subquadratic space requirement. We generalize the Johnson-Lindenstrauss Lemma to define “low-quality” mappings to a Euclidean space of significantly lower dimension, such that they satisfy a requirement weaker than approximately preserving all distances or even preserving the nearest neighbor. This mapping guarantees, with high probability, that an approximate nearest neighbor lies among the k approximate nearest neighbors in the projected space. These can be efficiently retrieved while using only linear storage by a data structure, such as BBD-trees. Our overall algorithm, given n points in dimension d, achieves space usage in O(dn), preprocessing time in O(dn logn), and query time in O(dn logn), where ρ is proportional to 1− 1/log logn, for fixed ∈ (0, 1). The dimension reduction is larger if one assumes that pointsets possess some structure, namely bounded expansion rate. We implement our method and present experimental results in up to 500 dimensions and 106 points, which show that the practical performance is better than predicted by the theoretical analysis. In addition, we compare our approach with E2LSH. 1998 ACM Subject Classification F.2.2 [Analysis of algorithms and problem complexity] Geometrical problems and computations