*Update*: [This recent paper][1] on this topic may also be of interest; it's quite short and claims to have a fully constructive approach.

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I think the closest to answering your question is the following paper.

"Column subset selection, matrix factorization, and eigenvalue optimization", by J. A. Tropp.
*In Proc. 2009 ACM-SIAM Symp. Discrete Algorithms (SODA), pp. 978-986, New York, NY, Jan. 2009.*  [SODA .pdf][2] or a [longer arXiv version][3].

From the abstract:

> Most research from the algorithms and numerical linear algebra communities focuses on a variant called rank-revealing QR, which seeks a well-conditioned collection of columns that spans the (numerical) range of the matrix. 
>
>....
>
> a celebrated result of Bourgain and Tzafriri demonstrates that each matrix with normalized columns contains a large column submatrix that is exceptionally well conditioned. Unfortunately, standard proofs of this result cannot be regarded as algorithmic. This paper presents a randomized, polynomial-time algorithm that produces the submatrix promised by Bourgain and Tzafriri.


  [1]: http://arxiv.org/abs/1509.00748
  [2]: http://users.cms.caltech.edu/~jtropp/conf/Tro09-Column-Subset-SODA.pdf
  [3]: https://arxiv.org/abs/0806.4404