Abstract
A Local Linear Embedding (LLE) module enhances the performance of two Evolutionary Computation (EC) algorithms employed as search tools in global optimization problems. The LLE employs the stochastic sampling of the data space inherent in Evolutionary Computation in order to reconstruct an approximate mapping from the data space back into the parameter space. This allows to map the target data vector directly into the parameter space in order to obtain a rough estimate of the global optimum, which is then added to the EC generation. This process is iterated and considerably improves the EC convergence. Thirteen standard test functions and two real-world optimization problems serve to benchmark the performance of the method. In most of our tests, optimization aided by the LLE mapping outperforms standard implementations of a genetic algorithm and a particle swarm optimization. The number and ranges of functions we tested suggest that the proposed algorithm can be considered as a valid alternative to traditional EC tools in more general applications. The performance improvement in the early stage of the convergence also suggests that this hybrid implementation could be successful as an initial global search to select candidates for subsequent local optimization.
References
Balasubramanian, M., Schwartz, E.L., Tenenbaum, J.B., de Silva, V., Langford, J.C.: The isomap algorithm and topological stability. Science 295(5552), 7a (2002)
Boschetti, F., Dentith, M., List, R.: Inversion of seismic refraction data using genetic algorithms. Geophysics, 1715–1727 (1996)
Boschetti, F., Dentith, M., List, R.: Inversion of gravity and magnetic data by genetic algorithm. Geophys. Prospect. 461–478 (1997)
Boschetti, F., Moresi, L.: Interactive inversion in geosciences. Geophysics 64, 1226–1235 (2001)
Cox, T., Cox, M.: Multidimensional Scaling. Chapman & Hall, London (1994)
Davis, L.: Handbook on Genetic Algorithms. Van Nostrand Reinhold (1991)
Donoho, D.L., Grimes, C.E.: Hessian eigenmaps: locally linear embedding techniques for highdimensional data. Proc. Nat. Acad. Arts Sci. 100, 5591–5596 (2003)
Jolliffe, I.T.: Principal Component Analysis. Springer, New York, 1986, 271 p
Goldberg, D.E.: Genetic Algorithms in Search, Optimization, and Machine Learning. Addison-Wesley Inc. (1989)
Kouropteva, O., Okun, O., Hadid, A., Soriano, M., Marcos, S., Pietikäinen, M.: Beyond locally linear embedding algorithm. Technical Report MVG-01-2002, University of Oulu, Machine Vision Group, Information Processing Laboratory (2002)
Moré, J.J., Garbow, B.S., Hillstrom, K.E.: Testing unconstrained optimization software. ACM Trans. Math. Softw. 7, 17–41 (1981)
Mouser, C., Dunn, S.: Comparing genetic algorithms and particle swarm optimisation for an inverse problem exercise. In: The 12th Biennial Computational Techniques and Applications Conference, Melbourne, Australia, 2004 (submitted)
Oldenburg, D.W., Ellis, R.G.: Inversion of geophysical data using an approximate inverse mapping. Geophys. J. Int. 105, 325–353 (1991)
Pohlheim, H.: GEATbx: Genetic and Evolutionary Algorithm Toolbox for use with MATLAB (2005). http://www.geatbx.com/docu/index.html
Roweis, S.T., Saul, L.K.: Nonlinear dimensionality reduction by locally linear embedding. Science 290, 2323–2326 (2000)
Saul, L., Roweis, S.: Think globally, fit locally: unsupervised learning of nonlinear manifolds. Technical Report MS CIS-02-18, University of Pennsylvania (2002), 37 p
Tenenbaum, J.B., de Silva, V., Langford, J.C.: A global geometric framework for nonlinear dimensionality reduction. Science 290, 2319–2323 (2000)
Wijns, C., Boschetti, F., Moresi, L.: Inversion in geology by interactive evolutionary computation. J. Struct. Geol. 25(10), 1615–1621 (2003). doi:10.1016/S0191-8141(03)00010-5
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Boschetti, F. A local linear embedding module for evolutionary computation optimization. J Heuristics 14, 95–116 (2008). https://doi.org/10.1007/s10732-007-9030-6
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10732-007-9030-6