Batch-Orthogonal Locality-Sensitive Hashing for Angular Similarity

Sign-random-projection locality-sensitive hashing (SRP-LSH) is a widely used hashing method, which provides an unbiased estimate of pairwise angular similarity, yet may suffer from its large estimation variance. We propose in this work batch-orthogonal locality-sensitive hashing (BOLSH), as a significant improvement of SRP-LSH. Instead of independent random projections, BOLSH makes use of batch-orthogonalized random projections, i.e., we divide random projection vectors into several batches and orthogonalize the vectors in each batch respectively. These batch-orthogonalized random projections partition the data space into regular regions, and thus provide a more accurate estimator. We prove theoretically that BOLSH still provides an unbiased estimate of pairwise angular similarity, with a smaller variance for any angle in (0,π), compared with SRP-LSH. Furthermore, we give a lower bound on the reduction of variance. The extensive experiments on real data well validate that with the same length of binary code, BOLSH may achieve significant mean squared error reduction in estimating pairwise angular similarity. Moreover, BOLSH shows the superiority in extensive approximate nearest neighbor (ANN) retrieval experiments.