Proportional asymptotics of piecewise exponential proportional hazards models

We study the flexible piecewise exponential model in a high dimensional setting where the number of covariates $p$ grows proportionally to the number of observations $n$ and under the hypothesis of random uncorrelated Gaussian designs. We prove rigorously that the optimal ridge penalized log-likelihood of the model converges in probability to the saddle point of a surrogate objective function. The technique of proof is the Convex Gaussian Min-Max theorem of Thrampoulidis, Oymak and Hassibi. An important consequence of this result, is that we can study the impact of the ridge regularization on the estimates of the parameter of the model and the prediction error as a function of the ratio $p/n > 0$. Furthermore, these results represent a first step toward rigorously proving the (conjectured) correctness of several results obtained with the heuristic replica method for the Cox semi-parametric model.