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An Elementary Approach to the Problem of Column Selection in a Rectangular Matrix

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Part of the Lecture Notes in Computer Science book series (LNISA,volume 10710)


The problem of extracting a well conditioned submatrix from any rectangular matrix (with e.g. normalized columns) has been a subject of extensive research with applications to machine learning (rank revealing factorization, sparse solutions to least squares regression problems, clustering, \(\cdots \)), optimisation (low stretch spanning trees, \(\cdots \)), and is also connected with problems in functional and harmonic analysis (Bourgain-Tzafriri restricted invertibility problem).

In this paper, we provide a deterministic algorithm which extracts a submatrix \(X_S\) from any matrix X with guaranteed individual lower and upper bounds on each singular value of \(X_S\). We are also able to deduce a slightly weaker (up to a \(\log \)) version of the Bourgain-Tzafriri theorem as an immediate side result.

We end the paper with a description of how our method applies to the analysis of a large data set and how its numerical efficiency compares with the method of Spieman and Srivastava.


  • Bourgain Tzafriri theorem
  • Restricted invertibility
  • Column selection problems

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Chrétien, S., Darses, S. (2018). An Elementary Approach to the Problem of Column Selection in a Rectangular Matrix. In: Nicosia, G., Pardalos, P., Giuffrida, G., Umeton, R. (eds) Machine Learning, Optimization, and Big Data. MOD 2017. Lecture Notes in Computer Science(), vol 10710. Springer, Cham.

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-72925-1

  • Online ISBN: 978-3-319-72926-8

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