Machine Learning

, Volume 89, Issue 1, pp 67-86

First online:

Generating feature spaces for linear algorithms with regularized sparse kernel slow feature analysis

  • Wendelin BöhmerAffiliated withNeural Information Processing Group, Technische Universität Berlin Email author 
  • , Steffen GrünewälderAffiliated withCentre for Computational Statistics and Machine Learning, University College London
  • , Hannes NickischAffiliated withPhilips Research Laboratories
  • , Klaus ObermayerAffiliated withNeural Information Processing Group, Technische Universität Berlin


Without non-linear basis functions many problems can not be solved by linear algorithms. This article proposes a method to automatically construct such basis functions with slow feature analysis (SFA). Non-linear optimization of this unsupervised learning method generates an orthogonal basis on the unknown latent space for a given time series. In contrast to methods like PCA, SFA is thus well suited for techniques that make direct use of the latent space. Real-world time series can be complex, and current SFA algorithms are either not powerful enough or tend to over-fit. We make use of the kernel trick in combination with sparsification to develop a kernelized SFA algorithm which provides a powerful function class for large data sets. Sparsity is achieved by a novel matching pursuit approach that can be applied to other tasks as well. For small data sets, however, the kernel SFA approach leads to over-fitting and numerical instabilities. To enforce a stable solution, we introduce regularization to the SFA objective. We hypothesize that our algorithm generates a feature space that resembles a Fourier basis in the unknown space of latent variables underlying a given real-world time series. We evaluate this hypothesis at the example of a vowel classification task in comparison to sparse kernel PCA. Our results show excellent classification accuracy and demonstrate the superiority of kernel SFA over kernel PCA in encoding latent variables.


Time series Latent variables Unsupervised learning Slow feature analysis Sparse kernel methods Linear classification