Update: This recent paper on this topic may also be of interest; it's quite short and claims to have a fully constructive approach.
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 or a longer arXiv version.
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.
