We propose a constrained maximum partial likelihood estimator for dimension reduction in integrative (e.g., pan-cancer) survival analysis with high-dimensional covariates. We assume that for each population in the study, the hazard function follows a distinct Cox proportional hazards model. To borrow information across populations, we assume that all of the hazard functions depend only on a small number of linear combinations of the predictors. We estimate these linear combinations using an algorithm based on "distance-to-set" penalties. This allows us to impose both low-rankness and sparsity. We derive asymptotic results which reveal that our regression coefficient estimator is more efficient than fitting a separate proportional hazards model for each population. Numerical experiments suggest that our method outperforms related competitors under various data generating models. We use our method to perform a pan-cancer survival analysis relating protein expression to survival across 18 distinct cancer types. Our approach identifies six linear combinations, depending on only 20 proteins, which explain survival across the cancer types. Finally, we validate our fitted model on four external datasets and show that our estimated coefficients can lead to better prediction than popular competitors.

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