The space complexity of inner product filters

TL;DR

This paper investigates the minimal deterministic space complexity for inner product estimation between high-dimensional vectors, providing tight bounds and an improved upper bound for distinguishing inner products above or below certain thresholds.

Contribution

It establishes tight space complexity bounds for deterministic inner product estimation, improving previous bounds and handling the case where vectors are known or unknown.

Findings

Exact space bounds are characterized as $d \, \log_2(\frac{\sqrt{1-\beta}}{\varepsilon}) \pm \Theta(d)$ bits.

The upper bound is constructive and improves prior results by up to a factor of 2.

The lower bound applies even when one vector is known exactly, ensuring tightness of the bounds.

Abstract

Motivated by the problem of filtering candidate pairs in inner product similarity joins we study the following inner product estimation problem: Given parameters $d\in {\bf N}$, $\alpha>\beta\geq 0$ and unit vectors $x,y\in {\bf R}^{d}$ consider the task of distinguishing between the cases $\langle x, y\rangle\leq\beta$ and $\langle x, y\rangle\geq \alpha$ where $\langle x, y\rangle = \sum_{i=1}^d x_i y_i$ 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 $\langle x, y\rangle$ with an additive error bounded by $\varepsilon = \alpha - \beta$. We show that $d \log_2 \left(\tfrac{\sqrt{1-\beta}}{\varepsilon}\right) \pm \Theta(d)$ bits of information about each vector is necessary and sufficient. Our upper bound is constructive and improves a known upper bound of $d \log_2(1/\varepsilon) + O(d)$ by up to a factor of 2 when $\beta$ 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.