Abstract
This paper targets the specific issue of out-of-sample interpolation when mapping individuals to a learnt manifold. This process involves two successive interpolations, which we formulate by means of kernel functions: from the ambient space to the coordinates space parametrizing the manifold and reciprocally. We combine two existing interpolation schemes: (i) inexact matching, to take into account the data dispersion around the manifold, and (ii) a multiscale strategy, to overcome single kernel scale limitations. Experiments involve synthetic data, and real data from 108 subjects, representing myocardial motion patterns used for the comparison of individuals to both normality and to a given abnormal pattern, whose manifold representation has been learnt previously.
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References
Belkin, M., et al.: Manifold regularization: A geometric framework for learning from labeled and unlabeled examples. J. Mach. Learn. Res. 7, 2399–2434 (2006)
Bengio, Y., et al.: Out-of-sample extensions for LLE, isomap, MDS, eigenmaps, and spectral clustering. Adv. Neural. Inf. Process. Syst. 16, 177–184 (2004)
Bengio, Y., et al.: Representation learning: a review and new perspectives. arXiv:1206.5538v2 (2012)
Bermanis, A., et al.: Multiscale data sampling and function extension. Appl. Comput. Harmon. Anal. 34, 15–29 (2013)
Bruveris, M., et al.: Mixture of kernels and iterated semi direct product of diffeomorphisms groups. SIAM Multiscale Model Simul. 10, 1344–1368 (2012)
Coifman, R.R., Lafon, S.: Geometric harmonics: A novel tool for multiscale out-of-sample extension of empirical functions. Appl. Comput. Harmon. Anal. 21, 31–52 (2006)
De Craene, M., et al.: Spatiotemporal diffeomorphic free-form deformation: Application to motion and strain estimation from 3D echocardiography. Med. Image Anal. 16, 427–450 (2012)
Duchateau, N., et al.: A spatiotemporal statistical atlas of motion for the quantification of abnormalities in myocardial tissue velocities. Med. Image Anal. 15, 316–328 (2011)
Duchateau, N., et al.: Constrained manifold learning for the characterization of pathological deviations from normality. Med. Image Anal. 16, 1532–1549 (2012)
Etyngier, P., et al.: Projection onto a shape manifold for image segmentation with prior. In: Proc. IEEE Int. Conf. Image Process., vol. IV361–IV364 (2007)
Gerber, S., et al.: Manifold modeling for brain population analysis. Med. Image Anal. 14, 643–653 (2010)
Kwok, J.T.Y., Tsang, I.W.H.: The pre-image problem in kernel methods. IEEE Trans. Neural. Netw. 15, 1517–1525 (2004)
Parsai, C., et al.: Toward understanding response to cardiac resynchronization therapy: left ventricular dyssynchrony is only one of multiple mechanisms. Eur. Heart J. 30, 940–949 (2009)
Rabin, N., Coifman, R.R.: Heterogeneous datasets representation and learning using diffusion maps and laplacian pyramids. In: Proc. SIAM Int. Conf. Data Mining, pp. 189–199 (2012)
Tenenbaum, J.B., et al.: A global geometric framework for nonlinear dimensionality reduction. Science 290, 2319–2323 (2000)
Zhang, Z., et al.: Adaptive manifold learning. IEEE Trans. Pattern Anal. Mach. Intell. 34, 253–265 (2012)
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Duchateau, N., De Craene, M., Sitges, M., Caselles, V. (2013). Adaptation of Multiscale Function Extension to Inexact Matching: Application to the Mapping of Individuals to a Learnt Manifold. In: Nielsen, F., Barbaresco, F. (eds) Geometric Science of Information. GSI 2013. Lecture Notes in Computer Science, vol 8085. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-40020-9_64
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DOI: https://doi.org/10.1007/978-3-642-40020-9_64
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