Improving Random Projections With Extra Vectors to Approximate Inner Products

Abstract

This research concerns itself with increasing the accuracy of random projections used to quickly approximate the inner products of data vectors from a given dataset by adding additional information, namely, adding and storing more extra known vectors to the given dataset and associated information. We show how the variance of estimated inner products is reduced as more vectors are added, how variance reduction is related to the geometry of the dataset and moreover, the asymptotic behaviour of the variance as the number of extra vectors added goes to infinity. We provide the formulae governing the estimate of inner products for adding arbitrarily many extra vectors. Lastly, we demonstrate how to efficiently implement the computations of the estimates by showing we can use pre-computed and stored values for most of the computations. Numerical simulations are conducted to support the analytical results.