Least-Mean-Squares Coresets for Infinite Streams

Vladimir Braverman, Dan Feldman, Harry Lang, Daniela Rus, Adiel Statman

Research output: Contribution to journalArticlepeer-review

Abstract

Consider a stream of <inline-formula><tex-math notation="LaTeX">$d$</tex-math></inline-formula>-dimensional rows (points in <inline-formula><tex-math notation="LaTeX">$\mathbb {R}^{d}$</tex-math></inline-formula>) arriving sequentially. An <inline-formula><tex-math notation="LaTeX">$\epsilon$</tex-math></inline-formula>-coreset is a positively weighted subset that approximates their sum of squared distances to any linear subspace of <inline-formula><tex-math notation="LaTeX">$\mathbb{R}^{d}$</tex-math></inline-formula>, up to a <inline-formula><tex-math notation="LaTeX">$1 \pm \epsilon$</tex-math></inline-formula> factor. Unlike other data summarizations, such a coreset: (1) can be used to minimize faster any optimization function that uses this sum, such as regularized or constrained regression, (2) preserves input sparsity; (3) easily interpretable; (4) avoids numerical errors; (5) applies to problems with constraints on the input, such as subspaces that are spanned by few input points. Our main result is the first algorithm that returns such an <inline-formula><tex-math notation="LaTeX">$\epsilon$</tex-math></inline-formula>-coreset using finite and constant memory during the streaming, i.e., independent of <inline-formula><tex-math notation="LaTeX">$n$</tex-math></inline-formula>, the number of rows seen so far. The coreset consists of <inline-formula><tex-math notation="LaTeX">$O(d \log ^{2} d / \epsilon ^{2})$</tex-math></inline-formula> weighted rows, which is nearly optimal according to existing lower bounds of <inline-formula><tex-math notation="LaTeX">$\Omega (d / \epsilon ^{2})$</tex-math></inline-formula>. We support our findings with experiments on the Wikipedia dataset benchmarked against state-of-the-art algorithms.

Original languageEnglish
Pages (from-to)1-18
Number of pages18
JournalIEEE Transactions on Knowledge and Data Engineering
DOIs
StateAccepted/In press - 2022

Bibliographical note

Publisher Copyright:
IEEE

Keywords

  • Big Data
  • Big Data
  • Computational modeling
  • Coresets
  • Covariance matrices
  • Libraries
  • Memory management
  • Optimization
  • Random access memory
  • Sparse matrices
  • Streaming Algorithms

ASJC Scopus subject areas

  • Information Systems
  • Computer Science Applications
  • Computational Theory and Mathematics

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