We consider low-rank reconstruction of a matrix using its columns and we
present asymptotically optimal algorithms for both spectral norm and Frobenius
norm reconstruction. The main tools we introduce to obtain our r esults are:
(i) the use of fast approximate SVD-like decompositions for column
reconstruction, and (ii) two deter ministic algorithms for selecting rows from
matrices with orthonormal columns, building upon the sparse represen tation
theorem for decompositions of the identity that appeared in \cite{BSS09}.