  We consider the problem of subset selection for $\ell_{p}$ subspace
approximation, i.e., given $n$ points in $d$ dimensions, we need to pick a
small, representative subset of the given points such that its span gives
$(1+\epsilon)$ approximation to the best $k$-dimensional subspace that
minimizes the sum of $p$-th powers of distances of all the points to this
subspace. Sampling-based subset selection techniques require adaptive sampling
iterations with multiple passes over the data. Matrix sketching techniques give
a single-pass $(1+\epsilon)$ approximation for $\ell_{p}$ subspace
approximation but require additional passes for subset selection.
  In this work, we propose an MCMC algorithm to reduce the number of passes
required by previous subset selection algorithms based on adaptive sampling.
For $p=2$, our algorithm gives subset selection of nearly optimal size in only
$2$ passes, whereas the number of passes required in previous work depend on
$k$. Our algorithm picks a subset of size $\mathrm{poly}(k/\epsilon)$ that
gives $(1+\epsilon)$ approximation to the optimal subspace. The running time of
the algorithm is $nd + d~\mathrm{poly}(k/\epsilon)$. We extend our results to
the case when outliers are present in the datasets, and suggest a two pass
algorithm for the same. Our ideas also extend to give a reduction in the number
of passes required by adaptive sampling algorithms for $\ell_{p}$ subspace
approximation and subset selection, for $p \geq 2$.
