  Audio source separation is often achieved by estimating the magnitude
spectrogram of each source, and then applying a phase recovery (or spectrogram
inversion) algorithm to retrieve time-domain signals. Typically, spectrogram
inversion is treated as an optimization problem involving one or several terms
in order to promote estimates that comply with a consistency property, a mixing
constraint, and/or a target magnitude objective. Nonetheless, it is still
unclear which set of constraints and problem formulation is the most
appropriate in practice. In this paper, we design a general framework for
deriving spectrogram inversion algorithm, which is based on formulating
optimization problems by combining these objectives either as soft penalties or
hard constraints. We solve these by means of algorithms that perform
alternating projections on the subsets corresponding to each
objective/constraint. Our framework encompasses existing techniques from the
literature as well as novel algorithms. We investigate the potential of these
approaches for a speech enhancement task. In particular, one of our novel
algorithms outperforms other approaches in a realistic setting where the
magnitudes are estimated beforehand using a neural network.
