Generalized massive optimal data compression
  Data compression has become one of the cornerstones of modern astronomical
data analysis, with the vast majority of analyses compressing large raw
datasets down to a manageable number of informative summaries. In this paper we
provide a general procedure for optimally compressing $N$ data down to $n$
summary statistics, where $n$ is equal to the number of parameters of interest.
We show that compression to the score function -- the gradient of the
log-likelihood with respect to the parameters -- yields $n$ compressed
statistics that are optimal in the sense that they preserve the Fisher
information content of the data. Our method generalizes earlier work on linear
Karhunen-Lo\'{e}ve compression for Gaussian data whilst recovering both
lossless linear compression and quadratic estimation as special cases when they
are optimal. We give a unified treatment that also includes the general
non-Gaussian case as long as mild regularity conditions are satisfied,
producing optimal non-linear summary statistics when appropriate. As a worked
example, we derive explicitly the $n$ optimal compressed statistics for
Gaussian data in the general case where both the mean and covariance depend on
the parameters.
