K-polytopes: a superproblem of k-means

  • Yigit Oktar
  • Mehmet TurkanEmail author
Original Paper


It has already been proven that under certain circumstances dictionary learning for sparse representations is equivalent to conventional k-means clustering. Through additional modifications on sparse representations, it is possible to generalize the notion of centroids to higher orders. In a related algorithm which is called k-flats, q-dimensional flats have been considered as alternative central prototypes. In the proposed formulation of this paper, central prototypes are instead simplexes or even more general polytopes. Using higher-dimensional, nonconvex prototypes may alleviate the curse of dimensionality while also enabling to model nonlinearly distributed datasets successfully. The proposed framework in this study can further be applied in supervised settings flexibly through one-class learning and also in other nonlinear frameworks through kernels.


Sparse representations Block sparsity Simplexes Polytopes Clustering Machine learning 



Authors are grateful to Prof. Dr. Turker Ince for fruitful discussions, and for his constructive comments that greatly improved the manuscript.

Supplementary material

11760_2019_1469_MOESM1_ESM.pdf (263 kb)
Supplementary material 1 (pdf 263 KB)


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© Springer-Verlag London Ltd., part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Computer EngineeringIzmir University of EconomicsIzmirTurkey
  2. 2.Department of Electrical and Electronics EngineeringIzmir University of EconomicsIzmirTurkey

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