  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.
