Abstract
Manifold learning methods model high-dimensional data through low-dimensional manifolds embedded in the observed data space. This simplification implies that their are prone to trustworthiness and continuity errors. Generative Topographic Mapping (GTM) is one such manifold learning method for multivariate data clustering and visualization, defined within a probabilistic framework. In the original formulation, GTM is optimized by minimization of an error that is a function of Euclidean distances, making it vulnerable to the aforementioned errors, especially for datasets of convoluted geometry. Here, we modify GTM to penalize divergences between the Euclidean distances from the data points to the model prototypes and the corresponding geodesic distances along the manifold. Several experiments with artificial data show that this strategy improves the continuity and trustworthiness of the data representation generated by the model.
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References
Figueiredo, M.A.T., Jain, A.K.: Unsupervised learning of finite mixture models. IEEE Transactions on Pattern Analysis and Machine Intelligence 24(3), 381–396 (2002)
Bishop, C.M., Svensén, M., Williams, C.K.I.: The Generative Topographic Mapping. Neural Computation 10(1), 215–234 (1998)
Vellido, A.: Missing data imputation through GTM as a mixture of t-distributions. Neural Networks 19(10), 1624–1635 (2006)
Vellido, A., Lisboa, P.J.G., Vicente, D.: Robust analysis of MRS brain tumour data using t-GTM. Neurocomputing 69(7-9), 754–768 (2006)
Archambeau, C., Verleysen, M.: Manifold constrained finite Gaussian mixtures. In: Cabestany, J., Gonzalez Prieto, A., Sandoval, F. (eds.) IWANN 2005. LNCS, vol. 3512, pp. 820–828. Springer, Heidelberg (2005)
Tenenbaum, J.B., de Silva, V., Langford, J.C.: A global geometric framework for nonlinear dimensionality reduction. Science 290, 2319–2323 (2000)
Lee, J.A., Lendasse, A., Verleysen, M.: Curvilinear Distance Analysis versus Isomap. In: Proceedings of European Symposium on Artificial Neural Networks (ESANN), pp. 185–192 (2002)
Bernstein, M., de Silva, V., Langford, J., Tenenbaum, J.: Graph approximations to geodesics on embedded manifolds. Technical report, Stanford University, CA (2000)
Dijkstra, E.W.: A note on two problems in connection with graphs. Numerische Mathematik 1, 269–271 (1959)
Venna, J., Kaski, S.: Neighborhood preservation in nonlinear projection methods: An experimental study. In: Dorffner, G., Bischof, H., Hornik, K. (eds.) ICANN 2001. LNCS, vol. 2130, pp. 485–491. Springer, Heidelberg (2001)
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Cruz-Barbosa, R., Vellido, A. (2008). On the Improvement of the Mapping Trustworthiness and Continuity of a Manifold Learning Model. In: Fyfe, C., Kim, D., Lee, SY., Yin, H. (eds) Intelligent Data Engineering and Automated Learning – IDEAL 2008. IDEAL 2008. Lecture Notes in Computer Science, vol 5326. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-88906-9_34
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DOI: https://doi.org/10.1007/978-3-540-88906-9_34
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