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Near Optimal Linear Algebra in the Online and Sliding Window Models.

Vladimir Braverman1, Petros Drineas2, Cameron Musco3

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Summary
This summary is machine-generated.

This study introduces new randomized and deterministic algorithms for numerical linear algebra problems in the sliding window model. These methods efficiently handle streaming data, offering significant improvements for spectral approximation and low-rank matrix approximation.

Keywords:
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Area of Science:

  • Numerical Linear Algebra
  • Streaming Data Algorithms
  • Theoretical Computer Science

Background:

  • The sliding window model processes recent data streams, posing challenges for traditional linear algebra algorithms.
  • Existing smooth histogram frameworks are insufficient for many linear-algebraic problems in the row-arrival sliding window model.
  • Key problems include spectral norms, generalized regression, and low-rank approximation.

Purpose of the Study:

  • To develop efficient algorithms for numerical linear algebra in the sliding window model.
  • To address limitations of existing methods for spectral and low-rank approximation.
  • To establish connections between the sliding window and online models.

Main Methods:

  • Introduced a unified row-sampling framework for randomized algorithms.
  • Utilized 'reverse online' sampling distributions (leverage scores, ℓ1 sensitivities, Lewis weights).
  • Developed a deterministic framework using merge-and-reduce and online coresets.

Main Results:

  • Achieved nearly optimal space and input sparsity runtime for spectral approximation, low-rank approximation, and ℓ1-subspace embeddings.
  • Provided the first sample-optimal online algorithm for low-rank approximation/projection-cost preservation.
  • Delivered the first online algorithm for ℓ1-subspace embeddings.

Conclusions:

  • The proposed row-sampling framework effectively addresses numerical linear algebra in the sliding window model.
  • The study bridges the gap between sliding window and online models, yielding new deterministic algorithms.
  • The developed techniques offer significant advancements for processing streaming data in linear algebra.