Cox regression in the proportional regime: a statistical mechanics perspective

Nowadays, many data-sets are \say{high dimensional}, that is the number of recorded features is comparable to, or larger than, the number of observations.The proportional regime, where the number of features in a regression model $(p)$ is proportional to the sample size $(n)$, is extremely challenging because the consistency of the Maximum A Posteriori estimator is lost. This hinders the study of the asymptotical behaviour of MAP estimators by means of traditional mathematical approaches. In this thesis, I adopt the statistical physics approach to optimization in order to obtain an improved asymptotic theory for the Maximum Partial Likelihood Estimator (MPLE) in the proportional regime, by means of the Replica method. We also proposed a fast iterative algorithm for fitting of the Regularized Cox model, analogous to the TAP equations in statistical physics of disordered systems. Since the Replica method is not fully mathematically rigorous, I also devoted time to rigorously prove that the Replica method effectively leads to the correct result for a flexible parametric version of the Cox model, called Piece-Wise Exponential (PWE). This, together with evidence from numerical simulations, corroborates the Replica results. The asymptotic theory obtained by the Replica method depends on several unknown parameters of the data generating process. In this thesis, we moved the first steps into making the theory practically applicable, proposing and testing algorithms to estimate these unknowns.