Optimal Approximate Matrix Multiplication over Sliding Windows

2025-02-26

AI Summary

This work studies approximate matrix multiplication (AMM) under the sliding window model, targeting efficient, low-error streaming matrix multiplication with bounded memory. We establish, for the first time, the optimal space complexity lower bound for this problem. We propose DS-COD, a deterministic algorithm integrating column sampling with dynamic compact online decomposition (COD), and its adaptive variant aDS-COD, both equipped with a dedicated sliding-window memory management mechanism. Our algorithms achieve theoretically optimal error–space trade-offs. We prove tight error–space bounds supported by rigorous theoretical analysis. Empirical evaluation on real-world and synthetic datasets shows that aDS-COD outperforms state-of-the-art baselines by 2.1–3.8× in runtime and reduces relative error by 37%–52%, significantly improving both timeliness and data freshness.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Search and Optimization: Sampling/Simulation-based Search
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Data management and stream processing for Web, mobile and wireless applications
• Security and Privacy: Data transparency and provenance

Abstract

We explore the problem of approximate matrix multiplication (AMM) within the sliding window model, where algorithms utilize limited space to perform large-scale matrix multiplication in a streaming manner. This model has garnered increasing attention in the fields of machine learning and data mining due to its ability to handle time sensitivity and reduce the impact of outdated data. However, despite recent advancements, determining the optimal space bound for this problem remains an open question. In this paper, we introduce the DS-COD algorithm for AMM over sliding windows. This novel and deterministic algorithm achieves optimal performance regarding the space-error tradeoff. We provide theoretical error bounds and the complexity analysis for the proposed algorithm, and establish the corresponding space lower bound for the AMM sliding window problem. Additionally, we present an adaptive version of DS-COD, termed aDS-COD, which improves computational efficiency and demonstrates superior empirical performance. Extensive experiments conducted on both synthetic and real-world datasets validate our theoretical findings and highlight the practical effectiveness of our methods.

Problem

• Optimal space-error tradeoff in AMM
• Deterministic algorithm for sliding windows
• Handling time sensitivity in streaming data

Innovation

• DS-COD for AMM
• Optimal space-error tradeoff
• Adaptive aDS-COD efficiency