Improved Bounds on Access-Redundancy Tradeoffs in Quantized Linear Computations

TL;DR

This paper improves theoretical bounds on the tradeoffs between access complexity and redundancy in quantized linear computations, and explores approximate recovery methods that outperform previous exact schemes.

Contribution

It provides new impossibility results for access-redundancy tradeoffs and introduces approximate recovery schemes that surpass earlier exact methods.

Findings

Improved impossibility bounds for general and block construction cases.

Optimality of block constructions within their class.

Effective approximate schemes for ε=0.1 outperform exact schemes.

Abstract

Consider the problem of computing quantized linear functions with only a few queries. Formally, given $\mathbf{x}\in \mathbb{R}^k$, our goal is to encode $\mathbf{x}$ as $\mathbf{c} \in \mathbb{R}^n$, for $n > k$, so that for any $\mathbf{w} \in A^k$, $\mathbf{w}^T \mathbf{x}$ can be computed using at most $\ell$ queries to $\mathbf{c}$. Here, $A$ is some finite set; in this paper we focus on the case where $|A| = 2$. Prior work \emph{(Ramkumar, Raviv, and Tamo, Trans. IT, 2024)} has given constructions and established impossibility results for this problem. We give improved impossibility results, both for the general problem, and for the specific class of construction (block construction) presented in that work. The latter establishes that the block constructions of prior work are optimal within that class. We also initiate the study of \emph{approximate} recovery for this problem, where the goal is not to recover $\mathbf{w}^T \mathbf{x}$ exactly but rather to approximate it up to a parameter $\varepsilon > 0$. We give several constructions, and give constructions for $\varepsilon = 0.1$ that outperform our impossibility result for exact schemes.