Abstract
In our recent publication [1], we began with an understanding that many real-world applications of multi-objective optimization involve a large number (10 or more) of objectives but then, existing evolutionary multi-objective optimization (EMO) methods have primarily been applied to problems having smaller number of objectives (5 or less). After highlighting the major impediments in handling large number of objectives, we proposed a principal component analysis (PCA) based EMO procedure, for dimensionality reduction, whose efficacy was demonstrated by solving upto 50-objective optimization problems. Here, we are addressing the fact that, when the data points live on a non-linear manifold or that the data structure is non-gaussian, PCA which yields a smaller dimensional ’linear’ subspace may be ineffective in revealing the underlying dimensionality. To overcome this, we propose two new non-linear dimensionality reduction algorithms for evolutionary multi-objective optimization, namely C-PCA-NSGA-II and MVU-PCA-NSGA-II. While the former is based on the newly introduced correntropy PCA [2], the later implements maximum variance unfolding principle [3,4,5] in a novel way. We also establish the superiority of these new EMO procedures over the earlier PCA-based procedure, both in terms of accuracy and computational time, by solving upto 50-objective optimization problems.
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Deb, K., Saxena, D.: Searching for pareto-optimal solutions through dimensionality reduction for certain large-dimensional multi-objective optimization problems. In: IEEE Congress on Evolutionary Computation, pp. 3353–3360. IEEE Computer Society Press, Los Alamitos (2006)
Wu, J.X., Pokharel, P., Paiva, A., Principe, J.C.: Nonlinear component analysis based on correntropy. In: IEEE Congress on Evolutionary Computation, pp. 3517–3521. IEEE Computer Society Press, Los Alamitos (2006)
Weinberger, K Q, Saul, L K: Unsupervised learning of image manifolds by semidefinite programming. International Journal of Computer Vision 70(1), 77–90 (2006)
Weinberger, K.Q., Saul, L.K.: An introduction to nonlinear dimensionality reduction by maximum variance unfolding. In: In Proceedings of the National Conference on Artificial Intelligence (AAAI), Nectar paper, AAAI, Menlo Park (2006)
Weinberger, K Q, Sha, F., Saul, L.K.: Learning a kernel matrix for nonlinear dimensionality reduction. In: Proceedings of the Twenty First International Conference on Machine Learning (ICML), pp. 839–846 (2004)
Deb, K.: Multi-objective optimization using evolutionary algorithms. Wiley, Chichester (2001)
Deb, K., Agrawal, S., Pratap, A., Meyarivan, T.: A fast and elitist multi-objective genetic algorithm: NSGA-II. IEEE Transactions on Evolutionary Computation 6(2), 182–197 (2002)
Saul, L.K., Weinberger, K.Q., Ham, J.H., Sha, F., Lee, D.D.: Spectral methods for dimensionality reduction. In: Semisupervised Learning, MIT Press, Cambridge (to appear, 2006), (smdr_ssl06.pdf at http://www.cs.ucsd.edu/~saul/abstracts.html )
Tenenbaum, J.B., de Silva, V., Langford, J.C.: A global geometric framework for nonlinear dimensionality reduction. Science 290, 2319–2323 (2000)
Belkin, M., Niyogi, P.: Laplacian eigenmaps for dimensionality reduction and data representation. Neural Computation 15, 1373–1396 (2003)
Roweis, S., Saul, L.: Nonlinear dimensionality reduction by locally linear embedding. Science 290, 2323–2326 (2000)
Ham, J.H., Lee, D.D., Mika, S.: A kernel view of the dimensionality reduction of manifolds. Technical Reports TR-110, Max Planck Institute for Biological Cybernatics, Tübingen, Germany (2003)
Cox, T.F., Cox, M.A.A.: Multidimensional Scaling. Chapman and Hall, London (1994)
Taylor, J.S., Cristianini, N.: Kernel methods for pattern analysis. Cambridge University Press, Cambridge (2004)
Scholkopf, B., Smola, A.J.: Learning With Kernels. MIT Press, Cambridge (2002)
Scholkopf, B., Smola, A., Muller, K.R.: Nonlinear component analysis as a kernel eigenvalue problem. Neural Computation 10(5), 1299–1319 (1998)
Deb, K., Thiele, L., Laumanns, M., Zitzler, E.: Scalable test problems for evolutionary multi-objective optimization. In: Congress on Evolutionary Computation (CEC), vol. 1, pp. 825–830 (2002)
Liu, C.: Capitalize on dimensionality increasing techniques for improving face recognition grand challenge performance. IEEE Transactions on Pattern Analysis and Machine Intelligence 28(5), 725–737 (2006)
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Saxena, D.K., Deb, K. (2007). Non-linear Dimensionality Reduction Procedures for Certain Large-Dimensional Multi-objective Optimization Problems: Employing Correntropy and a Novel Maximum Variance Unfolding. In: Obayashi, S., Deb, K., Poloni, C., Hiroyasu, T., Murata, T. (eds) Evolutionary Multi-Criterion Optimization. EMO 2007. Lecture Notes in Computer Science, vol 4403. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-70928-2_58
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DOI: https://doi.org/10.1007/978-3-540-70928-2_58
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