Audio signal reconstruction consists in recovering sound signals from incomplete or degraded representations. This problem can be cast as an inverse problem. Such problems are frequently tackled with the help of optimization or machine learning strategies. In this thesis, we propose to change the cost function in inverse problems related to audio signal reconstruction. We mainly address the phase retrieval problem, which is common when manipulating audio spectrograms. A first line of work tackles the optimization of non-quadratic cost functions for phase retrieval. We study this problem in two contexts: audio signal reconstruction from a single spectrogram and source separation. We introduce a novel formulation of the problem with Bregman divergences, as well as algorithms for its resolution. A second line of work proposes to learn the cost function from a given dataset. This is done under the framework of unfolded neural networks, which are derived from iterative algorithms. We introduce a neural network based on the unfolding of the Alternating Direction Method of Multipliers, that includes learnable activation functions. We expose the relation between the learning of its parameters and the learning of the cost function for phase retrieval. We conduct numerical experiments for each of the proposed methods to evaluate their performance and their potential with audio signal reconstruction
