  This paper addresses how well we can recover a data matrix when only given a
few of its elements. We present a randomized algorithm that element-wise
sparsifies the data, retaining only a few its elements. Our new algorithm
independently samples the data using sampling probabilities that depend on both
the squares ($\ell_2$ sampling) and absolute values ($\ell_1$ sampling) of the
entries. We prove that the hybrid algorithm recovers a near-PCA reconstruction
of the data from a sublinear sample-size: hybrid-($\ell_1,\ell_2$) inherits the
$\ell_2$-ability to sample the important elements as well as the regularization
properties of $\ell_1$ sampling, and gives strictly better performance than
either $\ell_1$ or $\ell_2$ on their own. We also give a one-pass version of
our algorithm and show experiments to corroborate the theory.
