Scalable nonlinear Cox modeling via random Fourier features with analytic uncertainty.

Abstract

Background: The Cox proportional hazards model often fails to capture complex biomedical risk structures, such as U-shaped biomarker associations, due to its assumption of linearity between the log-hazard and covariates. While existing kernel-based generalizations offer the necessary flexibility, their [Formula: see text] computational complexity limits applicability in large-scale cohort studies. Furthermore, most non-linear machine learning methods lack closed-form analytical measures of uncertainty for individual predictions..

Methods: A novel Random Fourier Features-based Cox regression approach (RFF-Cox) is presented to model non-linear risk relationships within a scalable framework. By mapping stationary kernels into a finite-dimensional explicit feature space, the method reduces computational complexity to [Formula: see text]. Model parameters are estimated via the Newton-Raphson algorithm on a ridge-regularized partial likelihood, while the bandwidth parameter ([Formula: see text]) is automatically optimized using a marginal likelihood criterion based on the Laplace approximation. Uncertainty quantification is performed via the Fisher information matrix, with a multivariate Delta method propagating variance from both parameter estimation and baseline hazard estimation. A Taylor-expansion-based inference framework enables covariate-level hazard ratio estimation, Wald-type significance testing, and formal interaction detection. Performance was evaluated using controlled simulations and four real-world datasets with sample sizes ranging from 500 to 9,105.

Results: In simulation scenarios, RFF-Cox degenerated to classical Cox estimates under linearity (RMSE: 0.111 vs. 0.110) while demonstrating a marked accuracy advantage under non-linearity (RMSE: 0.137 vs. 0.314), including the recovery of U-shaped risk functions. The true SBP × Smoking interaction was detected (estimate -1.093, p < 0.001). Analytical 95% CI coverage reached 95.9% in the linear scenario and 81.9% in the non-linear scenario. In real-world applications, the model exhibited discriminatory power competitive with Random Survival Forests and XGBoost with favorable computational efficiency. IPCW-weighted calibration analyses on the METABRIC dataset yielded low Integrated Calibration Error (ICI < 0.05) across most time horizons, though calibration slope deviations were noted at longer follow-up periods. Moreover, uncertainty in individual predictions, quantified via analytical confidence intervals, varied meaningfully across risk groups.

Conclusions: RFF-Cox provides a practical survival analysis framework that degenerates to the classical Cox model under linearity while offering non-linear modelling capabilities, computational efficiency, and formal inference tools - including covariate-level interpretation and interaction detection - that are not readily available in tree-based or deep learning survival methods.