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AeroSketch: Near-Optimal Time Matrix Sketch Framework for Persistent, Sliding Window, and Distributed Streams

Hanyan Yin, Dongxie Wen, Jiajun Li, Zhewei Wei, Xiao Zhang, Peng Zhao, Zhi-Hua Zhou

TL;DR

A novel matrix sketching framework that leverages recent advances in randomized numerical linear algebra (RandNLA), AeroSketch achieves optimal communication and space costs while delivering near-optimal update time complexity across persistent, sliding window, and distributed streaming scenarios.

Abstract

Many real-world matrix datasets arrive as high-throughput vector streams, making it impractical to store or process them in their entirety. To enable real-time analytics under limited computational, memory, and communication resources, matrix sketching techniques have been developed over recent decades to provide compact approximations of such streaming data. Some algorithms have achieved optimal space and communication complexity. However, these approaches often require frequent time-consuming matrix factorization operations. In particular, under tight approximation error bounds, each matrix factorization computation incurs cubic time complexity, thereby limiting their update efficiency. In this paper, we introduce AeroSketch, a novel matrix sketching framework that leverages recent advances in randomized numerical linear algebra (RandNLA). AeroSketch achieves optimal communication and space costs while delivering near-optimal update time complexity (within logarithmic factors) across persistent, sliding window, and distributed streaming scenarios. Extensive experiments on both synthetic and real-world datasets demonstrate that AeroSketch consistently outperforms state-of-the-art methods in update throughput. In particular, under tight approximation error constraints, AeroSketch reduces the cubic time complexity to the quadratic level. Meanwhile, it maintains comparable approximation quality while retaining optimal communication and space costs.

AeroSketch: Near-Optimal Time Matrix Sketch Framework for Persistent, Sliding Window, and Distributed Streams

TL;DR

A novel matrix sketching framework that leverages recent advances in randomized numerical linear algebra (RandNLA), AeroSketch achieves optimal communication and space costs while delivering near-optimal update time complexity across persistent, sliding window, and distributed streaming scenarios.

Abstract

Many real-world matrix datasets arrive as high-throughput vector streams, making it impractical to store or process them in their entirety. To enable real-time analytics under limited computational, memory, and communication resources, matrix sketching techniques have been developed over recent decades to provide compact approximations of such streaming data. Some algorithms have achieved optimal space and communication complexity. However, these approaches often require frequent time-consuming matrix factorization operations. In particular, under tight approximation error bounds, each matrix factorization computation incurs cubic time complexity, thereby limiting their update efficiency. In this paper, we introduce AeroSketch, a novel matrix sketching framework that leverages recent advances in randomized numerical linear algebra (RandNLA). AeroSketch achieves optimal communication and space costs while delivering near-optimal update time complexity (within logarithmic factors) across persistent, sliding window, and distributed streaming scenarios. Extensive experiments on both synthetic and real-world datasets demonstrate that AeroSketch consistently outperforms state-of-the-art methods in update throughput. In particular, under tight approximation error constraints, AeroSketch reduces the cubic time complexity to the quadratic level. Meanwhile, it maintains comparable approximation quality while retaining optimal communication and space costs.
Paper Structure (51 sections, 13 theorems, 48 equations, 11 figures, 1 table, 12 algorithms)

This paper contains 51 sections, 13 theorems, 48 equations, 11 figures, 1 table, 12 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries and Related Work
  3. Problem Definition: Matrix Sketching
  4. Full Stream Matrix Sketch
  5. Matrix Sketching over Sliding Window
  6. (At-the-time) Persistent Matrix Sketching
  7. Approximate Matrix Multiplication over Sliding Window
  8. Tracking Distributed Matrix Sketch
  9. RandNLA Operators
  10. Power Iteration
  11. Simultaneous Iteration
  12. Frequent Directions
  13. Our Framework: AeroSketch
  14. AeroSketch Framework Description
  15. High-level idea
  16. ...and 36 more sections

Key Result

corollary 1

For Algorithm alg:power, if we set $k = \lceil\log_{2} d\rceil+1$, then the probability that the estimated top squared singular value $\hat{\sigma}_1^2$ exceeds half of the true value $\sigma_1^2 / 2$ is bounded by: Power Iteration requires only matrix-vector multiplications. The total computational time cost of Algorithm alg:power is $O(d \ell k)$. If we set $k = \lceil\log_2 d\rceil+1$, then th

Figures (11)

  • Figure 1: Directly replacing the SVD in Fast-DS-FD with RandNLA would introduce cumulative restoring norm errors as shown in Eq. \ref{['eq:norm-error']}, whereas our AeroSketch does not.
  • Figure 2: Success probability lower bound with respect to $d$.
  • Figure 3: Amortized update time vs. parameter $\varepsilon$.
  • Figure 4: Space/communication cost vs. parameter $\varepsilon$.
  • Figure 5: Empirical relative covariance errors vs. parameter $\varepsilon$.
  • ...and 6 more figures

Theorems & Definitions (16)

  • Remark
  • Remark
  • corollary 1: Probabilistic error bound and time complexity of Power Iteration, a simplified corollary of Theorem 4.1(a) in kuczynski1992estimating
  • theorem 1: Probabilistic error bounds and time complexity of Simultaneous Iteration, the main lemma proven in musco2015randomized
  • theorem 2
  • corollary 2
  • theorem 3
  • theorem 4
  • theorem 5
  • theorem 6
  • ...and 6 more