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One-Pass Additive-Error Subset Selection for 𝓁_p Subspace Approximation

Authors Amit Deshpande, Rameshwar Pratap

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Amit Deshpande
  • Microsoft Research, Bengaluru, India
Rameshwar Pratap
  • Indian Institute of Technology, Mandi, H.P., India

Acknowledgements

We would like to thank anonymous reviewers for their careful reading of our manuscript and their insightful comments and suggestions.

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Amit Deshpande and Rameshwar Pratap. One-Pass Additive-Error Subset Selection for 𝓁_p Subspace Approximation. In 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 229, pp. 51:1-51:14, Schloss Dagstuhl – Leibniz-Zentrum fΓΌr Informatik (2022)
https://doi.org/10.4230/LIPIcs.ICALP.2022.51

Abstract

We consider the problem of subset selection for 𝓁_p subspace approximation, that is, to efficiently find a small subset of data points such that solving the problem optimally for this subset gives a good approximation to solving the problem optimally for the original input. Previously known subset selection algorithms based on volume sampling and adaptive sampling [Deshpande and Varadarajan, 2007], for the general case of p ∈ [1, ∞), require multiple passes over the data. In this paper, we give a one-pass subset selection with an additive approximation guarantee for 𝓁_p subspace approximation, for any p ∈ [1, ∞). Earlier subset selection algorithms that give a one-pass multiplicative (1+Ξ΅) approximation work under the special cases. Cohen et al. [Michael B. Cohen et al., 2017] gives a one-pass subset section that offers multiplicative (1+Ξ΅) approximation guarantee for the special case of 𝓁₂ subspace approximation. Mahabadi et al. [Sepideh Mahabadi et al., 2020] gives a one-pass noisy subset selection with (1+Ξ΅) approximation guarantee for 𝓁_p subspace approximation when p ∈ {1, 2}. Our subset selection algorithm gives a weaker, additive approximation guarantee, but it works for any p ∈ [1, ∞).

Subject Classification

ACM Subject Classification
  • Theory of computation β†’ Streaming models
  • Mathematics of computing β†’ Dimensionality reduction
  • Computing methodologies β†’ Dimensionality reduction and manifold learning
  • Theory of computation β†’ Sketching and sampling
Keywords
  • Subspace approximation
  • streaming algorithms
  • low-rank approximation
  • adaptive sampling
  • volume sampling
  • subset selection

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