  Non-convex methods for linear inverse problems with low-dimensional models
have emerged as an alternative to convex techniques. We propose a theoretical
framework where both finite dimensional and infinite dimensional linear inverse
problems can be studied. We show how the size of the the basins of attraction
of the minimizers of such problems is linked with the number of available
measurements. This framework recovers known results about low-rank matrix
estimation and off-the-grid sparse spike estimation, and it provides new
results for Gaussian mixture estimation from linear measurements. keywords:
low-dimensional models, non-convex methods, low-rank matrix recovery,
off-the-grid sparse recovery, Gaussian mixture model estimation from linear
measurements.
