Optimal Approximate Matrix Multiplication over Sliding Windows

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the fields of machine learning and data mining due to its ability to handle time sensitivity and reduce the impact of outdated data.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.