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.

Cox 比例风险模型是健康科学中对事件时间数据建模最广泛使用的方法之一。由于 Cox 偏似然函数的简单性，许多机器学习算法将其用于生存数据。然而，由于审查数据的性质，当采用更复杂的正则化时，优化问题变得棘手，这在处理高维组学数据时是必要的。在本文中，我们证明了基于 Fenchel 对偶性的 Cox 损失函数的凸共轭函数的存在，并为基于原始形式的优化提供了替代框架。此外，对偶形式提出了一种有效的算法，用于解决具有审查生存结果的核学习问题。我们通过模拟数据集和皮肤黑色素瘤 TCGA 数据集说明了 Cox 部分似然损失的派生对偶形式在多个核学习问题中的性能和属性。