DeepSurv: Personalized Treatment Recommender System Using A Cox Proportional Hazards Deep Neural Network

Medical practitioners use survival models to explore and understand the relationships between patients' covariates (e.g. clinical and genetic features) and the effectiveness of various treatment options. Standard survival models like the linear Cox proportional hazards model require extensive feature engineering or prior medical knowledge to model treatment interaction at an individual level. While nonlinear survival methods, such as neural networks and survival forests, can inherently model these high-level interaction terms, they have yet to be shown as effective treatment recommender systems. We introduce DeepSurv, a Cox proportional hazards deep neural network and state-of-the-art survival method for modeling interactions between a patient's covariates and treatment effectiveness in order to provide personalized treatment recommendations. We perform a number of experiments training DeepSurv on simulated and real survival data. We demonstrate that DeepSurv performs as well as or better than other state-of-the-art survival models and validate that DeepSurv successfully models increasingly complex relationships between a patient's covariates and their risk of failure. We then show how DeepSurv models the relationship between a patient's features and effectiveness of different treatment options to show how DeepSurv can be used to provide individual treatment recommendations. Finally, we train DeepSurv on real clinical studies to demonstrate how it's personalized treatment recommendations would increase the survival time of a set of patients. The predictive and modeling capabilities of DeepSurv will enable medical researchers to use deep neural networks as a tool in their exploration, understanding, and prediction of the effects of a patient's characteristics on their risk of failure.

The embedding dimension of Laplacian eigenfunction maps

Any closed, connected Riemannian manifold $M$ can be smoothly embedded by its Laplacian eigenfunction maps into $\mathbb{R}^m$ for some $m$. We call the smallest such $m$ the maximal embedding dimension of $M$. We show that the maximal embedding dimension of $M$ is bounded from above by a constant depending only on the dimension of $M$, a lower bound for injectivity radius, a lower bound for Ricci curvature, and a volume bound. We interpret this result for the case of surfaces isometrically immersed in $\mathbb{R}^3$, showing that the maximal embedding dimension only depends on bounds for the Gaussian curvature, mean curvature, and surface area. Furthermore, we consider the relevance of these results for shape registration.