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.