Semi-supervised rough fuzzy Laplacian Eigenmaps for dimensionality reduction

  • Minghua Ma
  • Tingquan DengEmail author
  • Ning Wang
  • Yanmei Chen
Original Article


Laplacian Eigenmaps is a popular nonlinear dimensionality reduction technique and there exist various scenarios of its extensions. In this paper, a semi-supervised rough fuzzy Laplacian Eigenmaps (SSRFLE) approach is developed for dimensionality reduction of high dimensional hybrid data. In the proposed method, a set of semi-supervised fuzzy similarity granules are constructed to characterize the similarity between samples according to the principle that homogeneous samples have higher similarity degrees than heterogeneous samples. A neighborhood rough fuzzy set model of such fuzzy similarity granules is built to assess the degrees two samples belong to the same class. A Laplacian nearest neighborhood graph and a class-related neighborhood graph are constructed to characterize the topological structure between samples and between each sample and its prototype to ensure homogeneous samples being mapped closer to and more compact around the prototypes in a lower dimensional space. In view of the fact that different features bring out distinct impacts on performances of feature extraction and clustering, the significance of each feature is assessed by designing an information entropy measure and the weighted distance between samples is incorporated into the proposed technique. A series of simulation experiments on real world hybrid datasets are carried out. Experimental results show superior performance of the proposed method in classification accuracy and data visualization compared with other state of the art semi-supervised methods.


Laplacian Eigenmaps Dimensionality reduction Information entropy Significance of feature Neighborhood rough fuzzy sets 



This work was supported by the National Natural Science Foundation of China (11471001).


  1. 1.
    Abd El-Monsef ME, El-Gayar MA, Aqeel RM (2017) A comparison of three types of rough fuzzy sets based on two universal sets. Int J Mach Learn Cybern 8:343–353CrossRefGoogle Scholar
  2. 2.
    Abdel-Mannan O, Ben Hamza A, Youssef A (2007) Incremental hessian locally linear embedding algorithm. IEEE Int Sympo Signal Process Appl 1–4Google Scholar
  3. 3.
    Bartholomew DJ (1983) Principal components analysis probability, statistical optics, and data testing. Springer, Berlin, HeidelbergGoogle Scholar
  4. 4.
    Belkin M, Niyogi P (2003) Laplacian Eigenmaps for dimensionality reduction and data representation. Neural Comput 15:1373–1396CrossRefzbMATHGoogle Scholar
  5. 5.
    Cai X, Wen G, Wei J, Li J, Yu Z (2014) Perceptual relativity-based semi-supervised dimensionality reduction algorithm. Appl Soft Comput 16:112–123CrossRefGoogle Scholar
  6. 6.
    Chen C, Zhang L, Bu J, Wang C, Chen W (2010) Constrained Laplacian Eigenmap for dimensionality reduction. Neurocomputing 73:951–958CrossRefGoogle Scholar
  7. 7.
    Costa JA, Hero AO (2004) Geodesic entropic graphs for dimension and entropy estimation in manifold learning. IEEE T Signal Process 52:2210–2221MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Costa JA, Hero AO (2005) Classification constrained dimensionality reduction. Proceedings of (ICASSP ’05). IEEE Int Conf Acoust Speech Signal Process 5:1077–1080Google Scholar
  9. 9.
    Deng TQ, Chen YM, Xu WL, Dai QH (2007) A novel approach to fuzzy rough sets based on a fuzzy covering. Inf Sci 177:2308–2326MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Dubois D, Prade H (1990) Rough fuzzy sets and fuzzy rough sets. Int J Gen Syst 17:191–209CrossRefzbMATHGoogle Scholar
  11. 11.
    Estévez PA, Tesmer M, Perez CA, Zurada J (2009) Normalized mutual information feature selection. IEEE T Neural Network 20:189–201CrossRefGoogle Scholar
  12. 12.
    Greco S, Matarazzo B, Slowinski R (2002) Rough approximation by dominance relations. Int J Intell Syst 17:153–171CrossRefzbMATHGoogle Scholar
  13. 13.
    Huang S, Zhuang L (2016) Exponential discriminant locality preserving projection for face recognition. Neurocomputing 208:373–377CrossRefGoogle Scholar
  14. 14.
    Hsu CC, Huang WH (2016) Integrated dimensionality reduction technique for mixed-type data involving categorical values. Appl Soft Comput 43:199–209CrossRefGoogle Scholar
  15. 15.
    Jȧrvinen J, Radeleczki S (2014) Rough sets determined by tolerances. Int J Approx Reason 55:1419–1438MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Jiang Q, Jia M, Hu J (2009) Machinery fault diagnosis using supervised manifold learning. Mech Syst Signal Process 23:2301–2311CrossRefGoogle Scholar
  17. 17.
    Keyhanian S, Nasersharif B (2014) Laplacian Eigenmaps modification using adaptive graph for pattern recognition. Int Sympo Telecommun 25–29Google Scholar
  18. 18.
    Kim K, Lee J (2014) Sentiment visualization and classification via semi-supervised nonlinear dimensionality reduction. Pattern Recogn 47:758–768CrossRefGoogle Scholar
  19. 19.
    Lai ZH, Wong WK, Xu Y, Yang J, Zhang D (2016) Approximate orthogonal sparse embedding for dimensionality reduction. IEEE T Neural Net Learn 27:723–735MathSciNetCrossRefGoogle Scholar
  20. 20.
    Li R (2013) A new supervised Laplacian Eigenmap for expression recognition. J Inf Comput Sci 10:4445–4451CrossRefGoogle Scholar
  21. 21.
    Li K, Kwong S (2014) A general framework for evolutionary multiobjective optimization via manifold learning. Neurocomputing 146:65–74CrossRefGoogle Scholar
  22. 22.
    Li WT, Xu WH (2015) Double-quantitative decision-theoretic rough set. Inf Sci 316:54–67MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Lichman M (2013) UCI machine learning repository. University of California, School of Information and Computer Science, Irvine, CA.
  24. 24.
    Lin RS, Yang MH, Levinson SE (2004) Object tracking using incremental Fisher discriminant analysis. Int Conf Pattern Recogn 2:757–760Google Scholar
  25. 25.
    Liu F, Zhang W, Gu S (2016) Local linear Laplacian Eigenmaps: a direct extension of LLE. Pattern Recogn Lett 75:30–35CrossRefGoogle Scholar
  26. 26.
    Malik ZK, Hussain A, Wu J (2016) An online generalized eigenvalue version of Laplacian Eigenmaps for visual big data. Neurocomputing 173:127–136CrossRefGoogle Scholar
  27. 27.
    Pawlak Z (1982) Rough sets. Int J Comput Inf Sci 11:341–356CrossRefzbMATHGoogle Scholar
  28. 28.
    Pollesch NL, Dale VH (2016) Normalization in sustainability assessment: methods and implications. Ecol Econ 130:195–208CrossRefGoogle Scholar
  29. 29.
    Raducanu B, Dornaika F (2012) A supervised non-linear dimensionality reduction approach for manifold learning. Pattern Recogn 45:2432–2444CrossRefzbMATHGoogle Scholar
  30. 30.
    Radzikowskaa AM, Kerre EE (2002) A comparative study of fuzzy rough sets. Fuzzy Sets Syst 126:137–155MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Roweis ST, Saul LK (2000) Nonlinear dimensionality reduction by locally linear embedding. Science 290:2323–2326CrossRefGoogle Scholar
  32. 32.
    Singer A (2006) Spectral independent component analysis. Appl Comput Harmon Anal 21:135–144MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Slowinski R, Vanderpooten D (2000) A generalized definition of rough approximations based on similarity. IEEE T Knowl Data En 12:331–336CrossRefGoogle Scholar
  34. 34.
    Susmaga R (2014) Reducts and constructs in classic and dominance-based rough sets approach. Inf Sci 277:45–54MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Suykens JA (2008) Data visualization and dimensionality reduction using kernel maps with a reference point. IEEE T Neural Network 19:1501–1517CrossRefGoogle Scholar
  36. 36.
    Tan A, Li J (2015) A kind of approximations of generalized rough set model. Int J Mach Learn Cybern 6:455–463CrossRefGoogle Scholar
  37. 37.
    Tenenbaum JB, De Silva V, Langford JC (2000) A global geometric framework for nonlinear dimensionality reduction. Science 290:2319–2323CrossRefGoogle Scholar
  38. 38.
    Thangavel K, Pethalakshmi A (2009) Dimensionality reduction based on rough set theory: a review. Appl Soft Comput 9:1–12CrossRefGoogle Scholar
  39. 39.
    Tsang ECC, Sun B, Ma W (2017) General relation-based variable precision rough fuzzy set. Int J Mach Learn Cybern 8:891–901CrossRefGoogle Scholar
  40. 40.
    Wang Q, Li J (2009) Combining local and global information for nonlinear dimensionality reduction. Neurocomputing 72:2235–2241CrossRefGoogle Scholar
  41. 41.
    Wang XZ, Dong CR, Fan TG (2007) Training T-S norm neural networks to refine weights for fuzzy if-then rules. Neurocomputing 70:2581–2587CrossRefGoogle Scholar
  42. 42.
    Wang XZ, Hong JR (1998) On the handling of fuzziness for continuous-valued attributes in decision tree generation. Fuzzy Sets Syst 99:283–290MathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    Wang XZ, Li CG (2005) A new definition of sensitivity for RBFNN and its applications to feature reduction. Lect Notes Comput Sci 3496:81–86CrossRefzbMATHGoogle Scholar
  44. 44.
    Wu WZ, Zhang WX (2002) Neighborhood operator systems and approximations. Inf Sci 144:201–217MathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    Xu WH, Guo YT (2016) Generalized multigranulation double-quantitative decision-theoretic rough set. Knowl Based Syst 105:190–205CrossRefGoogle Scholar
  46. 46.
    Xu J, Gu ZH, Xie K (2016) Fuzzy local mean discriminant analysis for dimensionality reduction. Neural Process Lett 44:701–718CrossRefGoogle Scholar
  47. 47.
    Xu W, Li WT (2016) Granular computing approach to two-way learning based on formal concept analysis in fuzzy datasets. IEEE Trans Cybern 46:366–379CrossRefGoogle Scholar
  48. 48.
    Xu J, Xie SL, Zhu WK (2017) Marginal patch alignment for dimensionality reduction. Soft Comput 21:2347–2356CrossRefGoogle Scholar
  49. 49.
    Yao YY (1998) Relational interpretations of neighborhood operators and rough set approximation. Inf Sci 111:239–259MathSciNetCrossRefzbMATHGoogle Scholar
  50. 50.
    Zadeh LA (1965) Fuzzy sets. Inf Control 8:338–353CrossRefzbMATHGoogle Scholar
  51. 51.
    Zhai J, Zhang Y, Zhu H (2017) Three-way decisions model based on tolerance rough fuzzy set. Int J Mach Learn Cybern 8:35–43CrossRefGoogle Scholar
  52. 52.
    Zhang Y, Li B, Wang W, Sun T, Yang X (2014) Supervised locally tangent space alignment for machine fault diagnosis. J Mech Sci Tech 28:2971–2977CrossRefGoogle Scholar
  53. 53.
    Zhu W (2007) Topological approaches to covering rough sets. Inf Sci 177:1499–1508MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.College of ScienceHarbin Engineering UniversityHarbinPeople’s Republic of China
  2. 2.Department of MathematicsHarbin Institute of TechnologyHarbinPeople’s Republic of China
  3. 3.Chengdu Aircraft Industrial (Group) Co., LtdChengduPeople’s Republic of China

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