We consider the efficient acquisition, parameter estimation, and recovery of signal ensembles that lie on a low-dimensional manifold in a high-dimensional ambient signal space. Our particular focus is on randomized, compressive acquisition of signals from the manifold generated by the transformation of a base signal by operators from a Lie group. Such manifolds factor prominently in a number of applications, including radar and sonar array processing, camera arrays, and video processing. Leveraging the fact that Lie group manifolds admit a convenient analytical characterization, we develop new theory and algorithms for: (1) estimating the Lie operator parameters from compressive measurements, and (2) recovering the base signal from compressive measurements. We validate our approach with several of numerical simulations, including the reconstruction of an affine-transformed video sequence from compressive measurements.