  We consider estimation of a deterministic unknown parameter vector in a
linear model with non-Gaussian noise. In the Gaussian case, dimensionality
reduction via a linear matched filter provides a simple low dimensional
sufficient statistic which can be easily communicated and/or stored for future
inference. Such a statistic is usually unknown in the general non-Gaussian
case. Instead, we propose a hybrid matched filter coupled with a randomized
compressed sensing procedure, which together create a low dimensional
statistic. We also derive a complementary algorithm for robust reconstruction
given this statistic. Our recovery method is based on the fast iterative
shrinkage and thresholding algorithm which is used for outlier rejection given
the compressed data. We demonstrate the advantages of the proposed framework
using synthetic simulations.
