A Scalable Nonparametric Continuous-Time Survival Model through Numerical Quadrature

TL;DR

QSurv is a scalable deep learning framework for nonparametric continuous-time survival analysis that uses numerical quadrature and time-conditioned low-rank adaptation to model complex hazard dynamics effectively.

Contribution

It introduces a novel training objective based on Gauss-Legendre quadrature and a time-conditioned low-rank adaptation mechanism for flexible, high-dimensional survival modeling.

Findings

QSurv achieves competitive predictive performance across diverse datasets.

It provides more interpretable hazard function estimates.

Theoretical bounds are established for cumulative-hazard approximation error.

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.