  In many real-world inverse problems, only incomplete measurement data are
available for training which can pose a problem for learning a reconstruction
function. Indeed, unsupervised learning using a fixed incomplete measurement
process is impossible in general, as there is no information in the nullspace
of the measurement operator. This limitation can be overcome by using
measurements from multiple operators. While this idea has been successfully
applied in various applications, a precise characterization of the conditions
for learning is still lacking. In this paper, we fill this gap by presenting
necessary and sufficient conditions for learning the underlying signal model
needed for reconstruction which indicate the interplay between the number of
distinct measurement operators, the number of measurements per operator, the
dimension of the model and the dimension of the signals. Furthermore, we
propose a novel and conceptually simple unsupervised learning loss which only
requires access to incomplete measurement data and achieves a performance on
par with supervised learning when the sufficient condition is verified. We
validate our theoretical bounds and demonstrate the advantages of the proposed
unsupervised loss compared to previous methods via a series of experiments on
various imaging inverse problems, such as accelerated magnetic resonance
imaging, compressed sensing and image inpainting.
