Improved Bounds on Access-Redundancy Tradeoffs in Quantized Linear Computations

2025-07-31

AI Summary

This paper studies low-query-complexity computation of quantized linear functions: given an input vector (x in mathbb{R}^k), encode it into a redundant representation (c in mathbb{R}^n) ((n > k)) such that, for any weight vector (w in A^k) where (A) is a size-2 finite set, the inner product (w^ op x) can be recovered either exactly or to (varepsilon)-accuracy using at most (ell) queries to (c). We propose the first unified framework supporting both exact and approximate recovery—breaking the information-theoretic lower bound of prior exact schemes. Leveraging tools from coding theory, combinatorial design, and quantized linear algebra, we construct novel encodings achieving superior trade-offs between query access and redundancy. We prove that classical block-based constructions are optimal within their class, and demonstrate—under (varepsilon = 0.1)—performance surpassing the fundamental limits of exact computation.

Technology Category

• Search and Optimization: Learning to Search
• Machine Learning: Quantum Machine Learning
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Querying, indexing, and retrieval in Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

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.

Problem

• Optimizing access-redundancy tradeoffs in quantized linear computations
• Improving impossibility results for exact recovery of linear functions
• Exploring approximate recovery schemes for quantized linear computations

Innovation

• Improved bounds on access-redundancy tradeoffs
• Optimal block constructions for quantized computations
• Approximate recovery schemes outperforming exact bounds