Probabilistic Polynomials and Hamming Nearest Neighbors

TL;DR

This paper introduces probabilistic polynomials for symmetric Boolean functions, enabling a subquadratic time algorithm for Hamming nearest neighbors in high dimensions, with applications to various similarity measures.

Contribution

It provides a constructive method for low-degree probabilistic polynomials for symmetric functions and applies this to develop the first subquadratic algorithm for Hamming nearest neighbors in high dimensions.

Findings

Probabilistic polynomials of degree O(√(n log(1/ε))) for symmetric Boolean functions.

First subquadratic time algorithm for batch Hamming distance in superlogarithmic dimensions.

Applications to maximum inner product, L1 closest pair, and Jaccard similarity.

Abstract

We show how to compute any symmetric Boolean function on $n$ variables over any field (as well as the integers) with a probabilistic polynomial of degree $O(\sqrt{n \log(1/\epsilon)})$ and error at most $\epsilon$. The degree dependence on $n$ and $\epsilon$ is optimal, matching a lower bound of Razborov (1987) and Smolensky (1987) for the MAJORITY function. The proof is constructive: a low-degree polynomial can be efficiently sampled from the distribution. This polynomial construction is combined with other algebraic ideas to give the first subquadratic time algorithm for computing a (worst-case) batch of Hamming distances in superlogarithmic dimensions, exactly. To illustrate, let $c(n) : \mathbb{N} \rightarrow \mathbb{N}$. Suppose we are given a database $D$ of $n$ vectors in $\{0,1\}^{c(n) \log n}$ and a collection of $n$ query vectors $Q$ in the same dimension. For all $u \in Q$, we wish to compute a $v \in D$ with minimum Hamming distance from $u$. We solve this problem in $n^{2-1/O(c(n) \log^2 c(n))}$ randomized time. Hence, the problem is in "truly subquadratic" time for $O(\log n)$ dimensions, and in subquadratic time for $d = o((\log^2 n)/(\log \log n)^2)$. We apply the algorithm to computing pairs with maximum inner product, closest pair in $\ell_1$ for vectors with bounded integer entries, and pairs with maximum Jaccard coefficients.