Fenchel duality of Cox partial likelihood with an application in survival kernel learning

The Cox proportional hazard model is one of the most widely used methods in modeling time-to-event data in the health sciences. Due to the simplicity of the Cox partial likelihood function, many machine learning algorithms use it for survival data. However, due to the nature of censored data, the optimization problem becomes intractable when more complicated regularization is employed, which is necessary when dealing with high dimensional omic data. In this paper, we show that a convex conjugate function of the Cox loss function based on Fenchel duality exists, and provide an alternative framework to optimization based on the primal form. Furthermore, the dual form suggests an efficient algorithm for solving the kernel learning problem with censored survival outcomes. We illustrate performance and properties of the derived duality form of Cox partial likelihood loss in multiple kernel learning problems with simulated and the Skin Cutaneous Melanoma TCGA datasets.