Abstract
In this paper, we propose a restorable autoencoder model as a non-linear method for reducing dimensionality. While non-linear methods can reduce the dimensionality of data more effectively than linear methods, they are, in general, not able to restore the original data from the dimensionality-reduced result. This is because non-linear methods provide a non-linear relationship between the original data and their dimensionality-reduced result. With the advantages of both linear and non-linear methods, the proposed model not only maintains an effective dimensionality reduction but also provides an observation-wise linear relationship with which the original data can be restored from the dimensionality-reduced result. We assessed the effectiveness of the proposed model and compared it with the linear method of principal component analysis and the non-linear methods of typical autoencoders using MNIST and Fashion-MNIST data sets. We demonstrated that the proposed model was more effective than or comparable to the compared methods in terms of the loss function and the reconstruction of input images. We also showed that the lower-dimensional projection obtained by the proposed model produced better or comparable classification results than that by the compared methods.
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W.M. Brown, S. Martin, S.N. Pollock, E.A. Coutsias, J. Watson, Algorithmic dimensionality reduction for molecular structure analysis. J. Chem. Phys. 129, 064118 (2008). https://doi.org/10.1063/1.2968610
M. Praprotnik, L. Delle Site, K. Kremer, Multiscale simulation of soft matter: from scale bridging to adaptive resolution. Annu. Rev. Phys. Chem. 59, 545 (2008). https://doi.org/10.1146/annurev.physchem.59.032607.093707
R. Everaers, M.R. Ejtehadi, Interaction potentials for soft and hard ellipsoids. Phys. Rev. E 67, 041710 (2003). https://doi.org/10.1103/PhysRevE.67.041710
D. Huang, H. Abdel-Khalik, C. Rabiti, F. Gleicher, Dimensionality reducibility for multi-physics reduced order modeling. Ann. Nucl. Energy 110, 526 (2017). https://doi.org/10.1016/j.anucene.2017.06.045
U. Kruger, J. Zhang, L. Xie, Developments and Applications of Nonlinear Principal Component Analysis: A Review, Edited by Gorban AN, Kégl B, Wunsch DC, Zinovyev AY (Springer, Berlin Heidelberg, 2008).
D. Donoho (200) High-dimensional data analysis: the curses and blessings of dimensionality (AMS Math Challenges Lecture. 2000), Chap. 1
J. Fan and R. Li (2006) In proceedings of the 25th international congress of mathematicians (Madrid, Spain, August 22–30, 2006)
J. Tenenbaum, V. Silva, J. Langford, A global geometric framework for nonlinear dimensionality reduction. Science 290, 2319 (2000). https://doi.org/10.1126/science.290.5500.2319
D. Rumelhart, G. Hinton, and R. Williams (1985) Learning internal representations by error propagation. Available from: https://app.dimensions.ai/details/publication/pub.1091744995. Accessed 12 Dec 2020
H. Abdi, L. Williams, Principal component analysis, WIREs. Comput. Stat. 2, 433 (2010). https://doi.org/10.1002/wics.101
M. Scholz, M. Fraunholz, and J. Selbig, Nonlinear principal component analysis: Neural network models and applications, edited by Gorban AN, Kégl B, Wunsch DC, Zinovyev AY (Springer, Berlin Heidelberg; 2008)
L. Van Der Maaten, E. Postma, J. Van den Herik, Dimensionality reduction: a comparative review. J. Mach. Learn. Res. 10, 66 (2009)
J. Cunningham, Z. Ghahramani, Linear dimensionality reduction: survey, insights, and generalizations. J. Mach. Learn. Res. 16, 2859 (2015)
S. Ladjal, A. Newson, and C. Pham, A PCA-like Autoencoder, ArXiv.abs/1904.01277 (2009).
M. Kramer, Nonlinear principal component analysis using autoassociative neural networks. AIChE. J. 37, 233 (1991). https://doi.org/10.1002/aic.690370209
D. Dong, T. McAvoy, Nonlinear principal component analysis-based on principal curves and neural networks. Comput. Chem. Eng. 20, 65 (1996). https://doi.org/10.1016/0098-1354(95)00003-K
R. Hahnloser, R. Sarpeshkar, M. Mahowald, R. Douglas, H. Seung, Digital selection and analogue amplification coexist in a cortex-inspired silicon circuit. Nature 405, 947 (2000). https://doi.org/10.1038/35016072
D. Erhan, Y. Bengio, A. Courville, P. Manzagol, P. Vincent, S. Bengio, Why does unsupervised pre-training help deep learning? J. Mach. Learn. Res. 11, 625 (2010)
G. Hinton, R. Salakhutdinov, Reducing the dimensionality of data with neural networks. Science 313, 504 (2006). https://doi.org/10.1126/science.1127647
G. Hinton, S. Osindero, Y. Teh, A fast learning algorithm for deep belief nets. Neural. Comput. 18, 1527 (2006). https://doi.org/10.1162/neco.2006.18.7.1527
Y. Bengio, P. Lamblin, D. Popovici, and H. Larochelle, Greedy Layer-Wise Training of Deep Networks, in Proceedings of the 19th International Conference on Neural Information Processing Systems. NIPS’06. (Cambridge, MA, USA: MIT Press, 2006)
L. Prechelt, In: Montavon G, Orr GB, Müller KR, editors. Early Stopping-But When? (Springer, Berlin Heidelberg, 2012). https://doi.org/10.1007/978-3-642-35289-8_5
C. Cortes, V. Vapnik, Support-vector networks. Mach. Learn. 20, 273 (1995). https://doi.org/10.1023/A:1022627411411
W. Greene, Econometric Analysis, 7th edn. (Pearson education, Boston, 2012).
M. Kuhn, A short introduction to the caret package. http://cran.r-project.org/web/packages/caret/vignettes/caret.pdf. Last accessed: 2020-02-12
F. Nielsen, Introduction to HPC with MPI for Data Science. Springer. Cham. (2016). https://doi.org/10.1007/978-3-319-21903-5_8
D. Meyer, Support vector machines. R. News. 1, 23 (2020)
C. Chang and C. Lin, LIBSVM: a library for support vector machines. http://www.csie.ntu.edu.tw/ cjlin/libsvm. Last accessed: 2020-12-12
https://www.rdocumentation.org/packages/nnet/versions/7.3-14. Last accessed: 2020-12-12
D. Powers, Evaluation: from precision, recall and F-measure to ROC, informedness, markedness and correlation. J. Mach. Learn. Technol. 2, 37 (2011)
K. Brodersen, C. Ong, K. Stephan, and J. Buhmann, The Balanced Accuracy and Its Posterior Distribution, in Proceedings of the 20th International Conference on Pattern Recognition (Istanbul, Turkey, August 23–26, 2010) 3121–3124
D. Joanes and C. Gill, comparing measures of sample Skewness and kurtosis. Journal of the Royal Statistical Society Series D (The Statistician) 47, 183 (1998). Available from: http://www.jstor.org/stable/2988433. Accessed 12 Dec 2020
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This work was supported by the research grant of the Kongju National University in 2020.
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Jeong, Y., Kim, S. & Lee, CY. A restorable autoencoder as a method for dimensionality reduction. J. Korean Phys. Soc. 78, 315–327 (2021). https://doi.org/10.1007/s40042-021-00074-6
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DOI: https://doi.org/10.1007/s40042-021-00074-6