natural images tend to be compressible, i.e., the amount of information needed to encode an image is small. this conciseness of information -- in other words, low dimensionality of the signal -- is found throughout a plethora of applications ranging from mri to quantum state tomography. it is natural to ask: can the number of measurements needed to determine a signal be comparable with the information content? we explore this question under modern models of low-dimensionality and measurement acquisition.