  Subset selection for the rank $k$ approximation of an $n\times d$ matrix $A$
offers improvements in the interpretability of matrices, as well as a variety
of computational savings. This problem is well-understood when the error
measure is the Frobenius norm, with various tight algorithms known even in
challenging models such as the online model, where an algorithm must select the
column subset irrevocably when the columns arrive one by one. In contrast, for
other matrix losses, optimal trade-offs between the subset size and
approximation quality have not been settled, even in the offline setting. We
give a number of results towards closing these gaps.
  In the offline setting, we achieve nearly optimal bicriteria algorithms in
two settings. First, we remove a $\sqrt k$ factor from a result of [SWZ19] when
the loss function is any entrywise loss with an approximate triangle inequality
and at least linear growth. Our result is tight for the $\ell_1$ loss. We give
a similar improvement for entrywise $\ell_p$ losses for $p>2$, improving a
previous distortion of $k^{1-1/p}$ to $k^{1/2-1/p}$. Our results come from a
technique which replaces the use of a well-conditioned basis with a slightly
larger spanning set for which any vector can be expressed as a linear
combination with small Euclidean norm. We show that this technique also gives
the first oblivious $\ell_p$ subspace embeddings for $1<p<2$ with $\tilde
O(d^{1/p})$ distortion, which is nearly optimal and closes a long line of work.
  In the online setting, we give the first online subset selection algorithm
for $\ell_p$ subspace approximation and entrywise $\ell_p$ low rank
approximation by implementing sensitivity sampling online, which is challenging
due to the sequential nature of sensitivity sampling. Our main technique is an
online algorithm for detecting when an approximately optimal subspace changes
substantially.
