  We consider a generalization of the classic linear regression problem to the
case when the loss is an Orlicz norm. An Orlicz norm is parameterized by a
non-negative convex function $G:\mathbb{R}_+\rightarrow\mathbb{R}_+$ with
$G(0)=0$: the Orlicz norm of a vector $x\in\mathbb{R}^n$ is defined as $
\|x\|_G=\inf\left\{\alpha>0\large\mid\sum_{i=1}^n G(|x_i|/\alpha)\leq
1\right\}. $ We consider the cases where the function $G(\cdot)$ grows
subquadratically. Our main result is based on a new oblivious embedding which
embeds the column space of a given matrix $A\in\mathbb{R}^{n\times d}$ with
Orlicz norm into a lower dimensional space with $\ell_2$ norm. Specifically, we
show how to efficiently find an embedding matrix $S\in\mathbb{R}^{m\times
n},m<n$ such that $\forall x\in\mathbb{R}^{d},\Omega(1/(d\log n)) \cdot
\|Ax\|_G\leq \|SAx\|_2\leq O(d^2\log n) \cdot \|Ax\|_G.$ By applying this
subspace embedding technique, we show an approximation algorithm for the
regression problem $\min_{x\in\mathbb{R}^d} \|Ax-b\|_G$, up to a $O(d\log^2 n)$
factor. As a further application of our techniques, we show how to also use
them to improve on the algorithm for the $\ell_p$ low rank matrix approximation
problem for $1\leq p<2$.
