publications

2024

1. VLDB 2024
   Best Paper
   Nomination

   

   Optimal Matrix Sketching over Sliding Windows

   Hanyan Yin, Dongxie Wen, Jiajun Li, Zhewei Wei^*, Xiao Zhang, Zengfeng Huang , and Feifei Li

   In VLDB 2024, Aug 2024

   Abs arXiv Bib PDF Code Poster Slides

   Matrix sketching, aimed at approximating a matrix \(\boldsymbol{A} ∈\mathbb{R}^{N \times d}\)consisting of vector streams of length \(N\)with a smaller sketching matrix \(\boldsymbol{B}∈\mathbb{R}^{\ell\times d}\), \(\ell ≪N\), 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 \(O\left({d\over \varepsilon}\right)\)space bound and provides a covariance error guarantee of \(\varepsilon=\lVert \boldsymbol{A}^⊤\boldsymbol{A}-\boldsymbol{B}^⊤\boldsymbol{B}\rVert_2/\lVert \boldsymbol{A}\rVert_F^2\). 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 \(O\left({d\over\varepsilon}\right)\)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 \(O\left({d\over\varepsilon}\right)\)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 DS-FD 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.

   @inproceedings{yin2024optimal,
     title = {Optimal Matrix Sketching over Sliding Windows},
     author = {Yin, Hanyan and Wen, Dongxie and Li, Jiajun and Wei, Zhewei and Zhang, Xiao and Huang, Zengfeng and Li, Feifei},
     booktitle = {VLDB 2024},
     dimensions = {true},
     doi = {10.14778/3665844.3665847},
     year = {2024},
     month = aug,
     volume = {17},
     issue = {9},
     pages = {2149--2161},
   }