Kernel methods performance in high dimensional phase retrieval: a statistical physics perspective

In this manuscript we exploit the powerful tools coming from statistical physics of disordered systems to characterize theoretically an high dimensional statistics problem, of great interest for the machine learning community. The main fuel of this work is the fact that, nonetheless artificial intelligence influences our everyday-life, being at the basis of numerous applications, many aspects of its theoretical foundation still are not perfectly understood. We work in this direction presenting theoretical predictions, found thanks to statistical physics techniques, for the training and generalization error in a supervised learning setting called in the literature phase retrieval. We study the performance of a specific class of learning algorithm based on the kernel method trying to highlight what is the role played by the distribution of the data which build the dataset in an high dimensional learning process. We study in detail when in this setting the learning performance of an equivalent model, called Gaussian Equivalent Model , could capture the learning curve of the experimental real-world one. Finally we analyze the overparametrized regime, shedding light on how classical statistics considerations can give us useful insights also if we work in an high dimensional regime.