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orthoMTL

Multi-task learning with orthogonal constraints for time-varying survival analysis

Overview

Standard survival models such as Cox proportional hazards assume that each feature’s effect is constant over time. orthoMTL relaxes this assumption by recasting survival analysis as a multi-task learning problem: binary classification tasks are defined at multiple time thresholds, and a shared coefficient matrix is estimated with an orthogonality penalty that encourages temporally distant tasks to use different features.

The method is based on:

Vervier, K., Mahé, P., d’Aspremont, A., Veyrieras, J.-B., & Vert, J.-P. (2014). On Learning Matrices with Orthogonal Columns or Disjoint Supports. ECML-PKDD 2014. https://hal.science/hal-00985654

The survival extension is described in:

Juric, D. et al. (2026). Genomic Determinants of Response to Alpelisib Plus Fulvestrant in the SOLAR-1 Trial. Annals of Oncology. https://doi.org/10.1016/j.annonc.2026.04.003

Installation

From CRAN (once released)

install.packages("orthoMTL")

Development version from GitHub

# install.packages("pak")
pak::pak("Novartis/orthoMTL")

Or using remotes:

# install.packages("remotes")
remotes::install_github("Novartis/orthoMTL")

Quick start

library(orthoMTL)

# Simulate data with known time-varying effects
sim <- simulate_mtl(n = 200, p = 20, n_signals = 5, seed = 42)

# Convert survival times to binary labels at each threshold
Y <- create_longitudinal_labels(sim$SurvTime, sim$Event, sim$thresholds)
W <- create_indicator_matrix(Y)   # censoring mask
K <- create_constraint_matrix(length(sim$thresholds))

# Fit the model
fit <- orthoMTL(sim$X, Y, lambda = 1e-3, K = K,
                survival = TRUE, censored.mat = W)
summary(fit)

# Visualise coefficients across time thresholds
plot_heatmap(fit)

Regression and classification

The same solver handles plain multi-task regression and classification — survival is just one of three modes. Select the mode on the solver with the logistic / survival flags, score with the matching metric, and (for CV) let cv_orthoMTL() pick the metric automatically.

library(orthoMTL)

# --- Multi-task regression ---
sim <- simulate_mtl(n = 200, p = 20, n_signals = 5,
                    mode = "regression", seed = 42)
fit <- orthoMTL(sim$X, sim$Y, lambda = 1e-3)      # logistic = survival = FALSE
pred <- predict(fit, newdata = sim$X)             # type = "link"
rmse_mtl(sim$Y, pred)
r2_mtl(sim$Y, pred)

# --- Multi-task classification (labels in {-1, +1}) ---
sim <- simulate_mtl(n = 200, p = 20, n_signals = 5,
                    mode = "classification", seed = 42)
fit <- orthoMTL(sim$X, sim$Y, lambda = 1e-3, logistic = TRUE)
prob <- predict(fit, newdata = sim$X, type = "response")   # P(Y = +1)
cls  <- predict(fit, newdata = sim$X, type = "class")      # {-1, +1}
auc_mtl(sim$Y, predict(fit, newdata = sim$X))              # AUC on link scores
accuracy_mtl(sim$Y, predict(fit, newdata = sim$X))

# CV picks the metric by mode (cindex / auc / rmse) unless overridden
cv <- cv_orthoMTL(sim$X, sim$Y, logistic = TRUE, survival = FALSE,
                  lambdas = c(1e-3, 1e-2), n_cores = 1)
cv$best

For a full walkthrough — including cross-validation, bootstrap inference, and comparison with Cox — see the package vignette:

vignette("introduction", package = "orthoMTL")

Key functions

Function	Purpose
`orthoMTL()`	Core solver (regression / classification / survival)
`cv_orthoMTL()`	Parallel hyperparameter grid search
`bootstrap_orthoMTL()`	Bootstrap coefficient stability vs null
`simulate_mtl()`	Simulate data with time-varying ground truth
`create_longitudinal_labels()`	Survival → binary label matrix
`create_indicator_matrix()`	Censoring indicator matrix
`create_constraint_matrix()`	Diffusion constraint matrix for tasks
`predict()`, `coef()`	Predictions (`type = "link"/"response"/"class"`) and coefficients
`cindex_mtl()`	Survival concordance index
`rmse_mtl()`, `r2_mtl()`	Regression metrics
`accuracy_mtl()`, `auc_mtl()`	Classification metrics
`plot_heatmap()`, `plot_correlation()`, `plot_bootstrap()`	Visualisation

Citation

If you use orthoMTL in your work, please cite:

@inproceedings{vervier2014,
  author    = {Vervier, Kevin and Mah{\'e}, Pierre and d'Aspremont,
               Alexandre and Veyrieras, Jean-Baptiste and Vert, Jean-Philippe},
  title     = {On Learning Matrices with Orthogonal Columns or Disjoint Supports},
  booktitle = {Machine Learning and Knowledge Discovery in Databases (ECML-PKDD 2014)},
  year      = {2014},
  publisher = {Springer},
  url       = {https://hal.science/hal-00985654}
}

License

GPL-3.

These binaries (installable software) and packages are in development.
They may not be fully stable and should be used with caution. We make no claims about them.
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