Local Density Estimation in High Dimensions

An important question that arises in the study of high-dimensional vector representations learned from data are, given a set $\mathcal{D}$ of vectors and a query q, estimate the number of points within a specified distance threshold of q. We develop two estimators, LSH count and multiprobe count that use locality-sensitive hashing to preprocess the data to accurately and efficiently estimate the answers to such questions via importance sampling. A key innovation is the ability to maintain a small number of hash tables via preprocessing data structures and algorithms that sample from multiple buckets in each hash table. We give bounds on the space requirements and sample complexity of our schemes and demonstrate their effectiveness in experiments on a standard word embedding data set.