Skip to main content
SpringerLink
Log in

A novel conformal deformation based sparse subspace clustering

  • Original Article
  • Published:
International Journal of Machine Learning and Cybernetics Aims and scope Submit manuscript

Abstract

Sparse Subspace Clustering (SSC) methods based on variant norms have attracted significant attention due to their empirical success. However, SSC methods presume that data points are embedded in a linear space and use the linear structure to find the sparse representation of data. Unfortunately, real-world datasets usually reside on a special manifold where linear geometry reduces the efficiency of clustering methods. This paper extends Sparse Subspace Clustering with \(l_1\)-norm and Entropy norm to cluster data points that lie in submanifolds of an unknown manifold. The key idea is to provide a novel feature space by conformal mapping the original intrinsic manifolds with unknown structures to n-spheres such that angles and sparse similarities of the original manifold data are preserved. The proposed method finds an appropriate distance instead of the Euclidean distance for the Kernel SSC algorithm and the SSC algorithm with Entropy norm. Finally, we provide the experimental analysis to compare the efficiency of stereographic sparse subspace clustering with \(l_1\)-norm, Entropy norm, and kernel on several data sets.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3

Similar content being viewed by others

Data Availability

The link of data is presented in the paper

Notes

  1. https://github.com/DeformationSSC/CDBSSC.

  2. https://github.com/sohanghosh29/Clustering-Codes.

  3. https://github.com/2232088201/Python-sparse-subspace-clustering-ADMM.

  4. http://vision.ucsd.edu/content/yale-face-database.

  5. http://featureselection.asu.edu/old/datasets.php.

  6. https://jundongl.github.io/scikit-feature/datasets.html.

  7. http://www.cad.zju.edu.cn/home/dengcai/Data/MLData.html.

  8. https://scikit-learn.org/stable/datasets/index.html.

References

  1. Aggarwal CC, Reddy CK (2014) Data clustering. Algorithms and applications. Chapman & Hall CRC Data mining and Knowledge Discovery series, Londra

  2. Baek S, Yoon G, Song J, Yoon SM (2021) Deep self-representative subspace clustering network. Pattern Recogn 118:108041

    Article  Google Scholar 

  3. Bai L, Liang J (2020) Sparse subspace clustering with entropy-norm. In: International Conference on Machine Learning. PMLR, pp 561–568

  4. Belkin M, Niyogi P (2002) Laplacian eigenmaps and spectral techniques for embedding and clustering. Adv Neural Inform Process Syst 14:585–591

    Google Scholar 

  5. Carmo MP (1992) Riemannian Geom. Birkhäuser

    Book  Google Scholar 

  6. Casselman B (2014) Stereographic projection, Feature column

  7. Chen Y, Li G, Gu Y (2017) Active orthogonal matching pursuit for sparse subspace clustering. IEEE Signal Process Lett 25(2):164–8

    Article  Google Scholar 

  8. Choi GP, Ho KT, Lui LM (2016) Spherical conformal parameterization of genus-0 point clouds for meshing. SIAM J Imag Sci 9(4):1582–618

    Article  MathSciNet  MATH  Google Scholar 

  9. Donoho D (2003) Hessian eigenmaps: new tools for nonlinear dimensionality reduction. Proc Natl Acad Sci 100:5591–5596

    Article  MATH  Google Scholar 

  10. Dyer EL, Sankaranarayanan AC, Baraniuk RG (2013) Greedy feature selection for subspace clustering. J Mach Learn Res 14(1):2487–517

    MathSciNet  MATH  Google Scholar 

  11. Elhamifar E, Vidal R (2013) Sparse subspace clustering: Algorithm, theory, and applications. IEEE Trans Pattern Anal Mach Intell 35(11):2765–81

    Article  Google Scholar 

  12. Favaro P, Vidal R, Ravichandran A (2011) A closed form solution to robust subspace estimation and clustering. In CVPR. IEEE, pp. 1801–1807

  13. Goh A, Vidal R (2008) Clustering and dimensionality reduction on Riemannian manifolds. In 2008 IEEE Conference on Computer Vision and Pattern Recognition, pp 1–7

  14. Jia H, Wang L, Song H, Mao Q, Ding S (2021) An efficient Nyström spectral clustering algorithm using incomplete Cholesky decomposition. Expert Syst Appl 186:115813

    Article  Google Scholar 

  15. Jing L, Ng MK, Xu J, Huang JZ (2005) Subspace clustering of text documents with feature weighting k-means algorithm. Pacific-Asia conference on knowledge discovery and data mining. Springer, Berlin, Heidelberg, pp 802–812

    Chapter  Google Scholar 

  16. Kanatani KI (2001) Motion segmentation by subspace separation and model selection. In Proceedings Eighth IEEE International Conference on computer Vision, pp. 586–591

  17. Kang Z, Lin Z, Zhu X, Xu W (2021) Structured graph learning for scalable subspace clustering: from single view to multiview. IEEE Transact Cybern

  18. Kang Z, Pan H, Hoi SC, Xu Z (2019) Robust graph learning from noisy data. IEEE Transact Cybern 50(5):1833–43

    Article  Google Scholar 

  19. Lee KC, Ho J, Kriegman DJ (2005) Acquiring linear subspaces for face recognition under variable lighting. IEEE Trans Pattern Anal Mach Intell 27(5):684–98

    Article  Google Scholar 

  20. Li P, Wang Q, Zuo W, Zhang L (2013) Log-Euclidean kernels for sparse representation and dictionary learning. In Proceedings of the IEEE international conference on computer vision, pp. 1601–1608

  21. Liu G, Lin Z, Yan S, Sun J, Yu Y, Ma Y (2012) Robust recovery of subspace structures by low-rank representation. IEEE Trans Pattern Anal Mach Intell 35(1):171–84

    Article  Google Scholar 

  22. Nguyen H, Yang W, Shen F, Sun C (2015) Kernel low-rank representation for face recognition. Neurocomputing 155:32–42

    Article  Google Scholar 

  23. Patel VM, Vidal R (2014) Kernel sparse subspace clustering. In 2014 ieee international conference on image processing (icip), pp. 2849–2853

  24. Pourkamali-Anaraki F, Folberth J, Becker S (2020) Efficient solvers for sparse subspace clustering. Signal Process 172:107548

    Article  Google Scholar 

  25. Qin Y, Zhang X, Shen L, Feng G (2022) Maximum Block Energy Guided Robust Subspace Clustering. IEEE Transactions on Pattern Analysis and Machine Intelligence

  26. Ratcliffe JG, Axler S, Ribet KA (2006) Foundations of hyperbolic manifolds. Springer, New York

    Google Scholar 

  27. Rao S, Tron R, Vidal R, Ma Y (2009) Motion segmentation in the presence of outlying, incomplete, or corrupted trajectories. IEEE Trans Pattern Anal Mach Intell 32(10):1832–45

    Article  Google Scholar 

  28. Roweis ST, Saul LK (2000) Nonlinear dimensionality reduction by locally linear embedding. Science 290(5500):2323–2326

    Article  Google Scholar 

  29. Toponogov VA (2006) Differential geometry of curves and surfaces. Birkhũser-Verlag, Basel

    Google Scholar 

  30. Tu LW (2011) An introduction to manifolds. Springer, New York

    Book  MATH  Google Scholar 

  31. Vidal R (2011) Subspace clustering. IEEE Signal Process Mag 28(2):52–68

    Article  Google Scholar 

  32. Wang YX, Xu H (2013) Noisy sparse subspace clustering. In International Conference on Machine Learning. PMLR, pp. 89–97

  33. Yang AY, Wright J, Ma Y, Sastry SS (2008) Unsupervised segmentation of natural images via lossy data compression. Comput Vis Image Underst 110(2):212–25

    Article  Google Scholar 

  34. Yin M, Guo Y, Gao J (2016) Kernel sparse subspace clustering on symmetric positive definite manifolds. In: Proceedings of the IEEE conference on computer vision and pattern recognition, pp 5157–5164

  35. You C, Robinson D, Vidal R (2016) Scalable sparse subspace clustering by orthogonal matching pursuit. In Proceedings of the IEEE conference on computer vision and pattern recognition, pp 3918–3927

  36. Zhang T, Szlam A, Lerman G (2009) Median k-flats for hybrid linear modeling with many outliers. In 2009 IEEE 12th International Conference on Computer Vision Workshops, ICCV Workshops. IEEE, pp 234–241

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Tarbiat Modares University, Tehran, Iran

    Kajal Eybpoosh & Abbas Heydari

  2. Department of Computer Science, Tarbiat Modares University, Tehran, Iran

    Mansoor Rezghi

Authors
  1. Kajal Eybpoosh
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Mansoor Rezghi
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Abbas Heydari
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Mansoor Rezghi.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Eybpoosh, K., Rezghi, M. & Heydari, A. A novel conformal deformation based sparse subspace clustering. Int. J. Mach. Learn. & Cyber. 14, 1579–1590 (2023). https://doi.org/10.1007/s13042-022-01712-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s13042-022-01712-6

Keywords

Search

Navigation