Journal of Mathematical Imaging and Vision

, Volume 49, Issue 1, pp 191–201 | Cite as

Multi-class Transductive Learning Based on 1 Relaxations of Cheeger Cut and Mumford-Shah-Potts Model

  • Xavier BressonEmail author
  • Xue-Cheng Tai
  • Tony F. Chan
  • Arthur Szlam


Recent advances in 1 optimization for imaging problems provide promising tools to solve the fundamental high-dimensional data classification in machine learning. In this paper, we extend the main result of Szlam and Bresson (Proceedings of the 27th International Conference on Machine Learning, pp. 1039–1046, 2010), which introduced an exact 1 relaxation of the Cheeger ratio cut problem for unsupervised data classification. The proposed extension deals with the multi-class transductive learning problem, which consists in learning several classes with a set of labels for each class. Learning several classes (i.e. more than two classes) simultaneously is generally a challenging problem, but the proposed method builds on strong results introduced in imaging to overcome the multi-class issue. Besides, the proposed multi-class transductive learning algorithms also benefit from recent fast 1 solvers, specifically designed for the total variation norm, which plays a central role in our approach. Finally, experiments demonstrate that the proposed 1 relaxation algorithms are more accurate and robust than standard 2 relaxation methods s.a. spectral clustering, particularly when considering a very small number of labels for each class to be classified. For instance, the mean error of classification for the benchmark MNIST dataset of 60,000 data in \(\mathbb{R}^{784}\) using the proposed 1 relaxation of the multi-class Cheeger cut is 2.4 % when only one label is considered for each class, while the error of classification for the 2 relaxation method of spectral clustering is 24.7 %.


Data analysis Clustering Transductive learning Multi-class Ratio cut Min cut Potts and Mumford-Shah energies Exact relaxation Fast L1 optimization Total variation 



Xavier Bresson is supported by the Hong Kong RGC under Grant GRF110311.


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Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • Xavier Bresson
    • 1
    Email author
  • Xue-Cheng Tai
    • 2
  • Tony F. Chan
    • 3
  • Arthur Szlam
    • 4
  1. 1.Department of Computer ScienceCity University of Hong KongKowloon TangHong Kong
  2. 2.Department of MathematicsUniversity of BergenBergenNorway
  3. 3.Department of Mathematics and Computer ScienceHong Kong University of Science and TechnologyKowloonHong Kong
  4. 4.Department of MathematicsThe City College of New YorkNew YorkUSA

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