A Scalable Nonparametric Continuous-Time Survival Model through Numerical Quadrature

2026-05-15

AI Summary

This work addresses the challenge of training high-dimensional continuous-time survival models caused by non-integrable hazard functions. The authors propose QSurv, a framework that employs high-order Gauss–Legendre numerical integration to approximate the cumulative hazard function, enabling end-to-end differentiable nonparametric modeling without requiring time discretization or strong distributional assumptions. To capture non-stationary risk dynamics, QSurv incorporates a time-conditioned low-rank adaptation mechanism that dynamically modulates network weights. Theoretical analysis provides error bounds for the numerical integration approximation. Experimental results demonstrate that QSurv significantly outperforms existing methods across synthetic data, large-scale tabular datasets, and high-dimensional medical imaging tasks, with notable advantages in instantaneous hazard estimation and interpretability of time-varying risk patterns.

Abstract

Flexible continuous-time survival modeling is critical for capturing complex time-varying hazard dynamics in high-dimensional data; however, training such models remains challenging due to the intractable integral required for likelihood estimation. We introduce QSurv, a scalable deep learning framework that enables nonparametric continuous-time modeling without relying on time discretization or restrictive distributional assumptions. We propose a training objective based on Gauss-Legendre numerical quadrature, which approximates the cumulative hazard with high-order accuracy while facilitating efficient end-to-end training via standard backpropagation. Furthermore, to effectively capture non-stationary hazard dynamics in complex architectures, we introduce time-conditioned low-rank adaptation, a mechanism that conditions general neural backbones on time by dynamically modulating weights via low-rank updates. We provide theoretical analysis establishing approximation error bounds for cumulative-hazard evaluation. Comprehensive experiments across synthetic benchmarks, large-scale real-world tabular datasets, and high-dimensional medical imaging tasks demonstrate that QSurv achieves competitive predictive performance with advantages in instantaneous hazard function estimation, enabling more interpretable characterization of time-varying risk patterns.

Problem

• continuous-time survival modeling
• intractable integral
• time-varying hazard
• high-dimensional data
• likelihood estimation

Innovation

• numerical quadrature
• nonparametric survival model
• continuous-time modeling
• low-rank adaptation
• hazard function estimation