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Modified Locally Linear Embedding with Affine Transformation

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Abstract

Dimensional reduction is a primary way to analyze and work with complex and large amount of multidimensional data by avoiding the effect of curse of dimensionality. This problem of constructing low dimensional embedding gains importance in number of fields like artificial intelligence, image processing, geographical research and lot more. In this paper, we introduce a modified locally linear embedding, an unsupervised learning algorithm that computes low dimensional data from complex high dimensional data using affine transformation and neighborhood preserving embedding. Unlike novel locally linear embedding, our method is affine invariant where each point is being represented by an affine combination of its neighboring points. At the end, we conduct the experiment to evaluate our proposed method and compare its performance with existing methods. Results show that our proposed method is unaffected by affine transformation, specifically shear while existing methods fail to produce correct results in case of shear.

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References

  1. Lawrence KS, Sam TR (2003) Think globally, fit locally: unsupervised learning of low dimensional manifolds. J Mach Learn Res 4:119–155

    MathSciNet  MATH  Google Scholar 

  2. John AL, Michel V (2007) Nonlinear dimensionality reduction. Springer, New York

  3. Shuicheng Y, Dong X, Benyu Z, Hong JZ, Qiang Y, Stephen L (2007) Graph embedding and extensions: a general framework for dimensionality reduction. IEEE Trans Pattern Anal Mach Intell 29(1):40–51

    Article  Google Scholar 

  4. Jolliffe IT (1986) Principal component analysis. Springer, New York

    Book  MATH  Google Scholar 

  5. Cox T, Cox M (1994) Multidimensional scaling. Chapman and Hall, London

    MATH  Google Scholar 

  6. Sam TR, Lawrence KS (2000) Nonlinear dimensionality reduction by locally linear embedding. Science 290(5500):2323–2326. doi:10.1126/science.290.5500.2323

    Article  Google Scholar 

  7. Tenenbaum JB, Silva V, Langford JC (2000) A global geometric framework for nonlinear dimensionality reduction. Science 290(5500):2319–2323

    Article  ADS  CAS  PubMed  Google Scholar 

  8. Belkin M, Niyogi P (2003) Laplacian eigenmaps for dimensionality reduction and representation. Neural Comput 15(6):1373–1396

    Article  MATH  Google Scholar 

  9. Donoho DL, Grimes G (2003) Hessian eigenmaps: locally linear embedding techniques for high dimensional data. Proc Natl Acad Sci 100(10):5591–5596

    Article  ADS  MathSciNet  CAS  PubMed  PubMed Central  MATH  Google Scholar 

  10. Zhen-yue Z, Hong-yuan Z (2004) Principal manifolds and nonlinear dimensionality reduction via tangent space alignment. J Shanghai Univ 8(4):406–424

    Article  MathSciNet  Google Scholar 

  11. Coifman RR, Lafon S, Lee AB, Maggioni M, Nadler B, Warner F, Zucker SW (2005) Geometric diffusion as a tool for harmonic analysis and structure definition of data: diffusion maps. Proc Natl Acad Sci 102(21):7426–7431

    Article  ADS  CAS  PubMed  PubMed Central  Google Scholar 

  12. Zhenyue Z, Wang J (2007) MLLE: modified locally linear embedding using multiple weights. In: Proceedings of advances in neural information processing systems, vol 19, pp 1593–1600

  13. Zhenyue Z, Jing W, Hongyuan Z (2012) Adaptive manifold learning. IEEE Trans Pattern Anal Mach Intell 34(2):253–265

    Article  Google Scholar 

  14. Yu S, Heng Q, Keqiu L, Yingwei J, Deqin Y, Shusheng G (2015) An effective discretization method for disposing high-dimensional data. Inf Sci 270:73–91

    Article  MathSciNet  MATH  Google Scholar 

  15. Guangbin W, Jun L, Yilin H, Qinyi C (2015) Fault diagnosis of supervision and homogenization distance based on local linear embedding algorithm. Math Probl Eng 2015:981598. doi:10.1155/2015/981598

  16. Zhihui L, Wai KW, Yong X, Jian Y, David Z (2015) Approximate orthogonal sparse embedding for dimensionality reduction. IEEE Trans Neural Netw Learn Syst 27(4):723–735. doi:10.1109/TNNLS.2015.2422994

    MathSciNet  Google Scholar 

  17. Xianglei X, Kejun W, Zhuowen L, Yu Z, Sidan D (2015) Fusion of local manifold learning methods. IEEE Signal Process Lett 22(4):395–399

    Article  ADS  Google Scholar 

  18. Dayong Z, Juan W (2015) Speech recognition using locality preserving projection based on multi kernel learning supervision. In: International symposium on computers and informatics, pp 1508–1516

  19. Modenov PS, Parkhomenko AS (2014) Euclidean and affine transformations: geometric transformations. Academic Press, New York

    MATH  Google Scholar 

  20. Tenenbaum JB (1998) Mapping a manifold of perceptual observations. In: Proceedings of advances in neural information processing systems, vol 10, pp 682–688

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Authors and Affiliations

  1. Department of Computer Science and Engineering, Jaypee University of Information Technology, Solan, India

    Kanika Mehta, Gajendra Tyagi, Aashish Rao & Pardeep Kumar

  2. GLA University, Mathura, India

    Durg Singh Chauhan

Authors
  1. Kanika Mehta
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  2. Gajendra Tyagi
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  3. Aashish Rao
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  4. Pardeep Kumar
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  5. Durg Singh Chauhan
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Corresponding author

Correspondence to Pardeep Kumar.

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Mehta, K., Tyagi, G., Rao, A. et al. Modified Locally Linear Embedding with Affine Transformation. Natl. Acad. Sci. Lett. 40, 189–196 (2017). https://doi.org/10.1007/s40009-017-0536-7

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  • DOI: https://doi.org/10.1007/s40009-017-0536-7

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