Abstract
We study data-adaptive dimensionality reduction in the context of supervised learning in general metric spaces. Our main statistical contribution is a generalization bound for Lipschitz functions in metric spaces that are doubling, or nearly doubling, which yields a new theoretical explanation for empirically reported improvements gained by preprocessing Euclidean data by PCA (Principal Components Analysis) prior to constructing a linear classifier. On the algorithmic front, we describe an analogue of PCA for metric spaces, namely an efficient procedure that approximates the data’s intrinsic dimension, which is often much lower than the ambient dimension. Our approach thus leverages the dual benefits of low dimensionality: (1) more efficient algorithms, e.g., for proximity search, and (2) more optimistic generalization bounds.
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References
Andoni, A., Krauthgamer, R.: The computational hardness of estimating edit distance. SIAM J. Comput. 39(6), 2398–2429 (2010)
Balcan, M.F., Blum, A., Vempala, S.: Kernels as features: On kernels, margins, and low-dimensional mappings. Mach. Learn. 65(1), 79–94 (2006)
Bartlett, P.L., Mendelson, S.: Rademacher and gaussian complexities: Risk bounds and structural results. JMLR 3, 463–482 (2002)
Bi, J., Bennett, K.P., Embrechts, M.J., Breneman, C.M., Song, M.: Dimensionality reduction via sparse support vector machines. JMLR 3, 1229–1243 (2003)
Blanchard, G., Zwald, L.: Finite-dimensional projection for classification and statistical learning. IEEE Trans. Inform. Theory 54(9), 4169–4182 (2008), http://dx.doi.org/10.1109/TIT.2008.926312
Blum, A.: Random projection, margins, kernels, and feature-selection. In: Saunders, C., Grobelnik, M., Gunn, S., Shawe-Taylor, J. (eds.) SLSFS 2005. LNCS, vol. 3940, pp. 52–68. Springer, Heidelberg (2006)
Burges, C.J.C.: Dimension reduction: A guided tour. Foundations and Trends in Machine Learning 2(4) (2010)
Der, R., Lee, D.: Large-Margin Classification in Banach Spaces. In: AISTATS 2007, pp. 91–98 (2007)
Enflo, P.: On the nonexistence of uniform homeomorphisms between L p -spaces. Ark. Mat. 8, 103–105 (1969)
Fukumizu, K., Bach, F.R., Jordan, M.I.: Dimensionality reduction for supervised learning with reproducing kernel hilbert spaces. JMLR 5, 73–99 (2004)
Golub, G.H., Van Loan, C.F.: Matrix computations, 3rd edn. Johns Hopkins University Press, Baltimore (1996)
Gottlieb, L.A., Kontorovich, A., Krauthgamer, R.: Adaptive metric dimensionality reduction (2013), http://arxiv.org/abs/1302.2752
Gottlieb, L.A., Kontorovich, L., Krauthgamer, R.: Efficient classification for metric data. In: COLT, pp. 433–440 (2010)
Gottlieb, L.A., Krauthgamer, R.: Proximity algorithms for nearly-doubling spaces. In: Serna, M., Shaltiel, R., Jansen, K., Rolim, J. (eds.) APPROX and RANDOM 2010. LNCS, vol. 6302, pp. 192–204. Springer, Heidelberg (2010)
Gupta, A., Krauthgamer, R., Lee, J.R.: Bounded geometries, fractals, and low-distortion embeddings. In: FOCS, pp. 534–543 (2003)
Hein, M., Bousquet, O., Schölkopf, B.: Maximal margin classification for metric spaces. J. Comput. Syst. Sci. 71(3), 333–359 (2005)
Huang, K., Aviyente, S.: Large margin dimension reduction for sparse image classification. In: SSP, pp. 773–777 (2007)
Koltchinskii, V., Panchenko, D.: Empirical margin distributions and bounding the generalization error of combined classifiers. Ann. Statist. 30(1), 1–50 (2002)
Kpotufe, S., Dasgupta, S.: A tree-based regressor that adapts to intrinsic dimension. J. Comput. Syst. Sci. 78(5), 1496–1515 (2012), http://dx.doi.org/10.1016/j.jcss.2012.01.002
Ledoux, M., Talagrand, M.: Probability in Banach Spaces. Springer (1991)
Lee, J.A., Verleysen, M.: Nonlinear Dimensionality Reduction. Information Science and Statistics. Springer (2007)
von Luxburg, U., Bousquet, O.: Distance-based classification with lipschitz functions. Journal of Machine Learning Research 5, 669–695 (2004)
Micchelli, C.A., Pontil, M.: A function representation for learning in banach spaces. In: Shawe-Taylor, J., Singer, Y. (eds.) COLT 2004. LNCS (LNAI), vol. 3120, pp. 255–269. Springer, Heidelberg (2004)
Mohri, M., Rostamizadeh, A., Talwalkar, A.: Foundations of Machine Learning. The MIT Press (2012)
Naor, A., Schechtman, G.: Planar earthmover is not in l 1. SIAM J. Comput. 37, 804–826 (2007)
Paul, S., Boutsidis, C., Magdon-Ismail, M., Drineas, P.: Random projections for support vector machines. CoRR abs/1211.6085 (2012)
Rahimi, A., Recht, B.: Random features for large-scale kernel machines. In: NIPS (2007)
Sabato, S., Srebro, N., Tishby, N.: Tight sample complexity of large-margin learning. In: NIPS, pp. 2038–2046 (2010)
Schölkopf, B., Shawe-Taylor, J., Smola, A., Williamson, R.: Kernel-dependent support vector error bounds. In: ICANN (1999)
Shawe-Taylor, J., Bartlett, P.L., Williamson, R.C., Anthony, M.: Structural risk minimization over data-dependent hierarchies. IEEE Transactions on Information Theory 44(5), 1926–1940 (1998)
Varshney, K.R., Willsky, A.S.: Linear dimensionality reduction for margin-based classification: High-dimensional data and sensor networks. IEEE Transactions on Signal Processing 59(6), 2496–2512 (2011)
Young, N.E.: Sequential and parallel algorithms for mixed packing and covering. In: FOCS, pp. 538–546 (2001)
Zhang, H., Xu, Y., Zhang, J.: Reproducing kernel banach spaces for machine learning. J. Mach. Learn. Res. 10, 2741–2775 (2009)
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Gottlieb, LA., Kontorovich, A., Krauthgamer, R. (2013). Adaptive Metric Dimensionality Reduction. In: Jain, S., Munos, R., Stephan, F., Zeugmann, T. (eds) Algorithmic Learning Theory. ALT 2013. Lecture Notes in Computer Science(), vol 8139. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-40935-6_20
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DOI: https://doi.org/10.1007/978-3-642-40935-6_20
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