Survival analysis is a crucial semi-supervised task in machine learning with numerous real-world applications, particularly in healthcare. Currently, the most common approach to survival analysis is based on Cox's partial likelihood, which can be interpreted as a ranking model optimized on a lower bound of the concordance index. This relation between ranking models and Cox's partial likelihood considers only pairwise comparisons. Recent work has developed differentiable sorting methods which relax this pairwise independence assumption, enabling the ranking of sets of samples. However, current differentiable sorting methods cannot account for censoring, a key factor in many real-world datasets. To address this limitation, we propose a novel method called Diffsurv. We extend differentiable sorting methods to handle censored tasks by predicting matrices of possible permutations that take into account the label uncertainty introduced by censored samples. We contrast this approach with methods derived from partial likelihood and ranking losses. Our experiments show that Diffsurv outperforms established baselines in various simulated and real-world risk prediction scenarios. Additionally, we demonstrate the benefits of the algorithmic supervision enabled by Diffsurv by presenting a novel method for top-k risk prediction that outperforms current methods.

Differentiable sorting algorithms allow training with sorting and ranking supervision, where only the ordering or ranking of samples is known. Various methods have been proposed to address this challenge, ranging from optimal transport-based differentiable Sinkhorn sorting algorithms to making classic sorting networks differentiable. One problem of current differentiable sorting methods is that they are non-monotonic. To address this issue, we propose a novel relaxation of conditional swap operations that guarantees monotonicity in differentiable sorting networks. We introduce a family of sigmoid functions and prove that they produce differentiable sorting networks that are monotonic. Monotonicity ensures that the gradients always have the correct sign, which is an advantage in gradient-based optimization. We demonstrate that monotonic differentiable sorting networks improve upon previous differentiable sorting methods. Recently, the idea of end-to-end training of neural networks with ordering supervision via continuous relaxation of the sorting function has been presented by Grover et al. [1]. The idea of ordering supervision is that the ground truth order of some samples is known while their absolute values remain unsupervised. This is done by integrating a sorting algorithm in the neural architecture.