Motivated by the problem of filtering candidate pairs in inner product similarity joins we study the following inner product estimation problem: Given parameters d∈ℕ, α>β≥0 and unit vectors x,y∈ ℝ^d consider the task of distinguishing between the cases ⟨x,y⟩≤β and ⟨x,y⟩≥α where ⟨x,y⟩ = ∑_{i=1}^d x_i y_i is the inner product of vectors x and y. The goal is to distinguish these cases based on information on each vector encoded independently in a bit string of the shortest length possible. In contrast to much work on compressing vectors using randomized dimensionality reduction, we seek to solve the problem deterministically, with no probability of error. Inner product estimation can be solved in general via estimating ⟨x,y⟩ with an additive error bounded by ε = α - β. We show that d log₂ (√{1-β}/ε) ± Θ(d) bits of information about each vector is necessary and sufficient. Our upper bound is constructive and improves a known upper bound of d log₂(1/ε) + O(d) by up to a factor of 2 when β is close to 1. The lower bound holds even in a stronger model where one of the vectors is known exactly, and an arbitrary estimation function is allowed.

Set similarity join is a fundamental and well-studied database operator. It is usually studied in the exact setting where the goal is to compute all pairs of sets that exceed a given similarity threshold (measured e.g. as Jaccard similarity). But set similarity join is often used in settings where 100% recall may not be important - indeed, where the exact set similarity join is itself only an approximation of the desired result set. We present a new randomized algorithm for set similarity join that can achieve any desired recall up to 100%, and show theoretically and empirically that it significantly improves on existing methods. The present state-of-the-art exact methods are based on prefix-filtering, the performance of which depends on the data set having many rare tokens. Our method is robust against the absence of such structure in the data. At 90% recall our algorithm is often more than an order of magnitude faster than state-of-the-art exact methods, depending on how well a data set lends itself to prefix filtering. Our experiments on benchmark data sets also show that the method is several times faster than comparable approximate methods. Our algorithm makes use of recent theoretical advances in high-dimensional sketching and indexing that we believe to be of wider relevance to the data engineering community.

A Bloom filter is a widely used data-structure for representing a set S and answering queries of the form “Is x in S?”. By allowing some false positive answers (saying ‘yes’ when the answer is in fact ‘no’) Bloom filters use space significantly below what is required for storing S. In the distance sensitive setting we work with a set S of (Hamming) vectors and seek a data structure that offers a similar trade-off, but answers queries of the form “Is x close to an element of S?” (in Hamming distance). Previous work on distance sensitive Bloom filters have accepted false positive and false negative answers. Absence of false negatives is of critical importance in many applications of Bloom filters, so it is natural to ask if this can be also achieved in the distance sensitive setting. Our main contributions are upper and lower bounds (that are tight in several cases) for space usage in the distance sensitive setting where false negatives are not allowed.