Abstract
Instead of classifying individual signals, we address classification of objects characterized by signal ensembles (i.e., collections of signals). Such necessity arises frequently in real situations: e.g., classification of video clips or object classification using acoustic scattering experiments to name a few. In particular, we propose an algorithm for classifying signal ensembles by bringing together well-known techniques from various disciplines in a novel way. Our algorithm first performs the dimensionality reduction on training ensembles using either the linear embeddings (e.g., Principal Component Analysis (PCA), Multidimensional Scaling (MDS)) or the nonlinear embeddings (e.g., the Laplacian eigenmap (LE), the diffusion map (DM)). After embedding training ensembles into a lower-dimensional space, our algorithm extends a given test ensemble into the trained embedding space, and then measures the “distance” between the test ensemble and each training ensemble in that space, and classify it using the nearest neighbor method. It turns out that the choice of this ensemble distance measure is critical, and our algorithm adopts the so-called Earth Mover’s Distance (EMD), a robust distance measure successfully used in image retrieval and image registration. We will demonstrate the performance of our algorithm using two real examples: classification of underwater objects using multiple sonar waveforms; and classification of video clips of digit-speaking lips. This article also provides a concise review on the several key concepts in statistical learning such as PCA, MDS, LE, DM, and EMD as well as the practical issues including how to tune parameters, which will be useful for the readers interested in numerical experiments.
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References
Basseville, M.: Distance measures for signal processing and pattern recognition. Signal Processing 18(4), 349–369 (1989)
Belkin, M., Niyogi, P.: Laplacian eigenmaps and spectral techniques for embedding and clustering. In: T.K. Leen, T.G. Dietterich, V. Tresp (eds.) Advances in Neural Information Processing Systems, vol. 13, pp. 585–591. The MIT Press, Cambridge, MA (2001)
Belkin, M., Niyogi, P.: Laplacian eigenmaps for dimensionality reduction and data representation. Neural Computation 15(6), 1373–1396 (2003)
Belkin, M., Niyogi, P.: Towards a theoretical foundation for Laplacian-based manifold methods. J. Comput. Syst. Sci. 74(8), 1289–1308 (2008)
Bengio, Y., Paiement, J.F., Vincent, P., Delalleau, O., Roux, N.L., Ouimet, M.: Out-of-sample extensions for LLE, Isomap, MDS, Eigenmaps, and spectral clustering. In: S. Thrun, L.K. Saul, B. Schölkopf (eds.) Advances in Neural Information Processing Systems, vol. 16, pp. 177–184. The MIT Press, Cambridge, MA (2004)
Bıyıkoğlu, T., Leydold, J., Stadler, P.F.: Laplacian Eigenvectors of Graphs, Lecture Notes in Mathematics, vol. 1915. Springer-Verlag, New York (2007)
Borg, I., Groenen, P.J.F.: Modern Multidimensional Scaling: Theory and Applications, 2nd edn. Springer, New York (2005)
Chung, F.R.K.: Spectral Graph Theory. No. 92 in CBMS Regional Conference Series in Mathematics. Amer. Math. Soc., Providence, RI (1997)
Coifman, R.R., Lafon, S.: Diffusion maps. Applied and Computational Harmonic Analysis 21(1), 5–30 (2006)
Coifman, R.R., Lafon, S.: Geometric harmonics. Applied and Computational Harmonic Analysis 21(1), 32–52 (2006)
Cover, T.M., Hart, P.: Nearest neighbor pattern classification. IEEE Trans. Inform. Theory IT-13, 21–27 (1967)
Donoho, D.L., Grimes, C.: Hessian eigenmaps: Locally linear embedding technique techniques for high-dimensional data. P. Natl. Acad. Sci. USA 100(10), 5591–5596 (2003)
Evans, L.C.: Partial differential equations and Monge-Kantorovich mass transfer. Tech. rep., Dept. Math., Univ. California, Berkeley (2001)
Fowlkes, C., Belongie, S., Chung, F., Malik, J.: Spectral grouping using the Nyström method. IEEE Trans. Pattern Anal. Machine Intell. 26(2), 214–225 (2004)
Gower, J.C.: Some distance properties of latent root and vector methods used in multivariate analysis. Biometrika 53(3/4), 325–338 (1966)
Haker, S., Zhu, L., Tannenbaum, A., Angenent, S.: Optimal mass transport for registration and warping. Intern. J. Comput. Vision 60(3), 225–240 (2004)
Hastie, T., Tibshirani, R., Friedman, J.: The Elements of Statistical Learning: Data Mining, Inference, and Prediction, 2nd edn. Springer-Verlag (2009)
Kirby, M., Sirovich, L.: Application of the Karhunen–Loève procedure for the characterization of human faces. IEEE Trans. Pattern Anal. Machine Intell. 12(1), 103–108 (1990)
Lafon, S., Keller, Y., Coifman, R.R.: Data fusion and multicue data matching by diffusion maps. IEEE Trans. Pattern Anal. Machine Intell. 28(11), 1784–1797 (2006)
Lafon, S., Lee, A.B.: Diffusion maps and coarse-graining: A unified framework for dimensionality reduction, graph partitioning and data set parameterization. IEEE Trans. Pattern Anal. Machine Intell. 28(9), 1393–1403 (2006)
Lafon, S.S.: Diffusion maps and geometric harmonics. Ph.D. thesis, Dept. Math., Yale Univ. (2004). Downloadable from http://www.math.yale.edu/~sl349
Lieu, L., Saito, N.: Automated discrimination of shapes in high dimensions. In: D. Van De Ville, V.K. Goyal, M. Papadakis (eds.) Wavelets XII, Proc. SPIE 6701 (2007). Paper # 67011V
von Luxburg, U.: A tutorial on spectral clustering. Stat. Comput. 17(4), 395–416 (2007)
van der Maaten, L.J.P., Postma, E.O., van den Herik, H.J.: Dimensionality reduction: A comparative review. Technical Report TiCC-TR 2009-005, Tilburg Centre for Creative Computing, Tilburg Univ. (2009)
Marchand, B.: Local signal analysis for classification. Ph.D. thesis, Dept. Math., Univ. California, Davis (2010)
Marchand, B., Saito, N., Xiao, H.: Classification of objects in synthetic aperture sonar images. In: Proc. 14th IEEE Workshop on Statistical Signal Processing, pp. 433–437. IEEE (2007)
Mardia, K.V., Kent, J.T., Bibby, J.M.: Multivariate Analysis. Academic Press, San Diego, CA (1979)
Murase, H., Nayar, S.K.: Visual learning and recognition of 3d objects from appearance. Intern. J. Comput. Vision 14(1), 5–24 (1995)
Nesbitt, C.L., Lopes, J.L.: Subcritical detection of an elongated target buried under a rippled interface. In: Oceans ’04, MTS/IEEE Techno-Ocean ’04, vol. 4, pp. 1945–1952 (2004)
Ng, A.Y., Jordan, M.I., Weiss, Y.: On spectral clustering: Analysis and an algorithm. In: T. Dietterich, S. Becker, Z. Ghahramani (eds.) Advances in Neural Information Processing Systems, vol. 14, pp. 849–856. The MIT Press, Cambridge, MA (2002)
Patterson, E.K., Gurbuz, S., Tufekci, Z., Gowdy, J.N.: Moving-talker, speaker-independent feature study, and baseline results using the CUAVE multimodal speech corpus. EURASIP J. Appl. Signal Process. 11, 1189–1201 (2002)
Roweis, S.T., Saul, L.K.: Nonlinear dimensionality reduction by locally linear embedding. Science 290(5500), 2323–2326 (2000)
Rubner, Y., Tomasi, C.: Perceptual Metrics for Image Database Navigation. Kluwer Academic Publishers, Boston, MA (1999)
Rubner, Y., Tomasi, C., Guibas, L.J.: The Earth Mover’s Distance as a metric for image retrieval. Intern. J. Comput. Vision 40(2), 99–121 (2000)
Saito, N.: Image approximation and modeling via least statistically dependent bases. Pattern Recognition 34, 1765–1784 (2001)
Saito, N., Coifman, R.R.: Local discriminant bases and their applications. J. Math. Imaging Vis. 5(4), 337–358 (1995). Invited paper
Saito, N., Coifman, R.R., Geshwind, F.B., Warner, F.: Discriminant feature extraction using empirical probability density estimation and a local basis library. Pattern Recognition 35(12), 2841–2852 (2002)
Saito, N., Woei, E.: Analysis of neuronal dendrite patterns using eigenvalues of graph Laplacians. JSIAM Letters 1, 13–16 (2009). Invited paper
Sanguinetti, G., Laidler, J., Lawrence, N.D.: Automatic determination of the number of clusters using spectral algorithms. In: Proc. 15th IEEE Workshop on Machine Learning for Signal Processing, pp. 55–60 (2005)
Zhang, X., Mersereau, R.M.: Lip feature extraction towards an automatic speechreading system. In: Proc. 2000 International Conference on Image Processing, vol. 3, pp. 226–229. IEEE (2000)
Zhang, Z., Zha, H.: Principal manifolds and nonlinear dimensionality reduction via tangent space alignment. SIAM J. Sci. Comput. 26(1), 313–338 (2005)
Zhou, S.K., Chellappa, R.: From sample similarity to ensemble similarity – Probabilistic distance measures in reproducing kernel Hilbert space. IEEE Trans. Pattern Anal. Machine Intell. 28(6), 917–929 (2006)
Acknowledgements
This work was partially supported by the ONR grants N00014-06-1-0615, N00014-07-1-0166, N00014-09-1-0041, N00014-09-1-0318, the NSF grant DMS-0410406, and the NSF VIGRE grants DMS-0135345, DMS-0636297. A preliminary version of a subset of the material in this article was presented at the SPIE Wavelets XII Conference, in San Diego, in August 2007 [ 22]. We thank Dr. Quyen Huynh and Dr. Joe Lopes of NSWC-PC for providing the experimental sonar data. We have used the SPECTRAL Toolbox version 0.1 distributed on the web by Dr. Guido Sanguinetti and Dr. Jonathan Laidler to compute Elongated Kmeans. Finally, we would like to thank Dr. Bradley Marchand for his help in processing the sonar data used in the numerical experiments and Mr. Julien van Hout for his help in numerical experiments on underwater object classification.
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Lieu, L., Saito, N. (2011). Signal Ensemble Classification Using Low-Dimensional Embeddings and Earth Mover’s Distance. In: Cohen, J., Zayed, A. (eds) Wavelets and Multiscale Analysis. Applied and Numerical Harmonic Analysis. Birkhäuser Boston. https://doi.org/10.1007/978-0-8176-8095-4_11
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