Matrix sketching, aimed at approximating a matrix $\boldsymbol{A} \in \mathbb{R}^{N\times d}$ consisting of vector streams of length $N$ with a smaller sketching matrix $\boldsymbol{B} \in \mathbb{R}^{\ell\times d}, \ell \ll N$, has garnered increasing attention in fields such as large-scale data analytics and machine learning. A well-known deterministic matrix sketching method is the Frequent Directions algorithm, which achieves the optimal $O\left(\frac{d}{\varepsilon}\right)$ space bound and provides a covariance error guarantee of $\varepsilon = \lVert \boldsymbol{A}^\top \boldsymbol{A} - \boldsymbol{B}^\top \boldsymbol{B} \rVert_2/\lVert \boldsymbol{A} \rVert_F^2$. The matrix sketching problem becomes particularly interesting in the context of sliding windows, where the goal is to approximate the matrix $\boldsymbol{A}_W$, formed by input vectors over the most recent $N$ time units. However, despite recent efforts, whether achieving the optimal $O\left(\frac{d}{\varepsilon}\right)$ space bound on sliding windows is possible has remained an open question. In this paper, we introduce the DS-FD algorithm, which achieves the optimal $O\left(\frac{d}{\varepsilon}\right)$ space bound for matrix sketching over row-normalized, sequence-based sliding windows. We also present matching upper and lower space bounds for time-based and unnormalized sliding windows, demonstrating the generality and optimality of \dsfd across various sliding window models. This conclusively answers the open question regarding the optimal space bound for matrix sketching over sliding windows. Furthermore, we conduct extensive experiments with both synthetic and real-world datasets, validating our theoretical claims and thus confirming the correctness and effectiveness of our algorithm, both theoretically and empirically.

翻译：矩阵素描旨在用更小的素描矩阵 $\boldsymbol{B} \in \mathbb{R}^{\ell\times d}, \ell \ll N$ 来近似由长度为 $N$ 的向量流组成的矩阵 $\boldsymbol{A} \in \mathbb{R}^{N\times d}$，在大规模数据分析和机器学习等领域日益受到关注。一种著名的确定性矩阵素描方法是 Frequent Directions 算法，它实现了最优的 $O\left(\frac{d}{\varepsilon}\right)$ 空间界，并提供了协方差误差保证 $\varepsilon = \lVert \boldsymbol{A}^\top \boldsymbol{A} - \boldsymbol{B}^\top \boldsymbol{B} \rVert_2/\lVert \boldsymbol{A} \rVert_F^2$。在滑动窗口场景下，矩阵素描问题变得尤为有趣，其目标是用输入向量在最近 $N$ 个时间单位内构成的矩阵 $\boldsymbol{A}_W$ 进行近似。然而，尽管已有近期研究，在滑动窗口上是否能够实现最优的 $O\left(\frac{d}{\varepsilon}\right)$ 空间界仍然是一个悬而未决的问题。在本文中，我们提出了 DS-FD 算法，该算法在基于行归一化和序列的滑动窗口上实现了矩阵素描的最优 $O\left(\frac{d}{\varepsilon}\right)$ 空间界。我们还给出了基于时间和非归一化滑动窗口的匹配上界和下界，展示了 \dsfd 在各种滑动窗口模型上的通用性和最优性。这最终回答了关于滑动窗口上矩阵素描最优空间界的开放问题。此外，我们使用合成数据集和真实数据集进行了大量实验，验证了我们的理论主张，从而从理论和实证两方面确认了我们算法的正确性和有效性。