SPaiK - Scalable Pairwise Kernel Learning Software

SPaiK is a scalable software package for pairwise kernel learning. It combines the stochastic inexact limited-memory bundle method (StoILMBM) for optimization, the stochastic generalized vec trick (sGVT) for efficient computation with pairwise Kronecker kernels, and a rich set of kernel functions provided by RLScore.

The included loss-functions for the pairwise kernel problem are:

• squared loss,
• squared epsilon-insensitive loss,
• epsilon-insensitive squared loss,
• epsilon-insensitive absolute loss,
• absolute loss.

Note that only the epsilon-insensitive squared loss has been tested for functionality.

Files included

• spaik.py

  • Main Python file. Includes RLScore calls.

• pkl_utility.py

  • Python utility programs.

• spaik.f95

  • Building block between Python and Fortran for pairwise learning software.

• slmba.f95

  • StoLMBM - a stochastic limited memory bundle method for nonsmooth optimization (specially developed for SPaiK).

• objfun.f95

  • Computation of the function and subgradients values with different loss functions. Selection between loss functions is made in spaik.py. Includes sGVT.

• initpkl.f95

  • Initialization of parameters and variables for SPaiK and StoLMBM. Includes modules:
    • initpkl - Initialization of parameters for SPaiK.
    • initslmba - Initialization of StoLMBM.

• parameters.f95

  • Parameters for Fortran. Inludes modules:
    • r_precision - Precision for reals,
    • param - Parameters.

• subpro.f95

  • Subprograms for StoLMBM.

• data.py

  • Contains functions to load the example datasets in SPaiK. The data files are assumed to be located in a folder "data". This repository does not include the datasets themselves; links to all example datasets are provided in repository github.com/TurkuML.

  • Contains functions to create train-test-validation splits. Splits are created for every experimental setting IDIT, IDOT, ODIT, and ODOT (see the references below).

• Makefile

  • makefile: Builds a shared library to allow StoLMBM (Fortran95 code) to be called from Python program SPaiK. Uses f2py, Python3.7, and requires a Fortran compiler (gfortran) to be installed.

Installation and usage

The source uses f2py and Python3.7, and requires a Fortran compiler (gfortran by default) and the RLScore to be installed.

To use the code:

1. Select the data and loss function from spaik.py file.
2. Run Makefile (by typing "make") to build a shared library that allows spaik.f95 (Fortran95 code) to be called from Python program spaik.py.
3. Finally, just type "python3.7 spaik.py".

The algorithm returns a csv-file with performance measures (C-index, IC-index, and MSE) computed in the test set under different experimental settings IDIT, IDOT, ODIT, and ODOT. The best results are selected using a separate validation set and validated w.r.t. C-index. In addition, separate csv-files with predictions under different experimental settings are returned.

References:

• SPaiK, sGVT, and StoILMBM:
  • N. Karmitsa, T. Pahikkala, A. Airola, "Scalable Pairwise Kernel Learning with Stochastic Vec Trick", ArXiv2606.16979, 2026.
• RLScore:
  • T. Pahikkala, A. Airola, "Rlscore: Regularized Least-Squares Learners", Journal of Machine Learning Research, Vol. 17, No. 221, pp. 1-5, 2016.
• InexactLMBM, LMBM, and SLMBA:
  • J. Lampainen, K. Joki, N. Karmitsa, and M.M. Mäkelä, "Inexact Limited Memory Bundle Method", ArXiv 2604.08067, 2026.
  • N. Haarala, K. Miettinen, M.M. Mäkelä, "Globally Convergent Limited Memory Bundle Method for Large-Scale Nonsmooth Optimization", Mathematical Programming, Vol. 109, No. 1, pp. 181-205, 2007.
  • M. Haarala, K. Miettinen, M.M. Mäkelä, "New Limited Memory Bundle Method for Large-Scale Nonsmooth Optimization", Optimization Methods and Software, Vol. 19, No. 6, pp. 673-692, 2004.
  • N. Karmitsa, V.-P. Eronen, M.M. Mäkelä, T. Pahikkala, A. Airola, "Stochastic Limited Memory Bundle Algorithm for Clustering in Big Data", Pattern Recognition, Vol 165, 111654, 2025. (A different version of the Stochastic LMBM).
• Generalized vec trick:
  • A. Airola, T. Pahikkala, "Fast Kronecker Product Kernel Methods via Generalized Vec Trick", IEEE Transactions on Neural Networks and Learning Systems, Vol. 29, pp. 3374–3387, 2018.
  • M. Viljanen, A. Airola, T. Pahikkala, "Generalized Vec Trick for Fast Learning of Pairwise Kernel Models", Machine Learning, Vol. 111, 543–573, 2022.
• Nonsmooth optimization:
  • A. Bagirov, N. Karmitsa, M.M. Mäkelä, "Introduction to Nonsmooth Optimization: Theory, Practice and Software", Springer , 2014.
• Interaction-Concordance-Index, data, and experimental settings:
  • T. Pahikkala, R. Numminen, P. Movahedi, N. Karmitsa, and A. Airola, "Interaction Concordance Index: Performance Evaluation for Interaction Prediction Methods", ArXiv2510.14419, 2025.

Acknowledgements

The work was financially supported by the Research Council of Finland, Project No. #340182 and #345804 led by Tapio Pahikkala and Project No. #340140 and #345805 led by Antti Airola.