Abstract
The previous chapter discussed the use of decision forests for estimating the latent density of unlabeled data. This has led to a forest-based probabilistic generative model which captures efficiently the “intrinsic” structure of the data themselves. The present chapter delves further into the issue of learning the structure of high dimensional data as well as mapping them onto a lower dimensional space, while preserving spatial relationships between data points. This task goes under the name of manifold learning and is closely related to dimensionality reduction and embedding.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
Multi-dimensional scaling (MDS) [73] or alternative techniques may also be considered.
- 2.
In practical applications the original space is usually of much higher dimensionality than 2D.
- 3.
If the input points were reordered correctly for each tree we would obtain an affinity matrix W t with a block-diagonal structure.
References
Belkin M, Niyogi P (2003) Laplacian eigenmaps for dimensionality reduction and data representation. Neural Comput
Belkin M, Niyogi P (2008) Towards a theoretical foundation for Laplacian-based manifold methods. J Comput Syst Sci 74(8)
Bishop CM, Svensen M, Williams CKI (1998) GTM: the generative topographic mapping. Neural Comput
Cayton L (2005) Algorithms for manifold learning. Technical Report CS2008-0923, University of California, San Diego
Chapelle O, Schölkopf B, Zien A (2006) Semi-supervised learning. MIT Press, Cambridge
Cox TF, Cox MAA (2001) Multidimensional scaling. Chapman and Hall, London
De Porte J, Herbst BM, Hereman W, van Der Walt SJ (2008) An introduction to diffusion maps. Techniques
Duchateau N, De Craene M, Piella G, Frangi AF (2011) Characterizing pathological deviations from normality using constrained manifold learning. In: Proc medical image computing and computer assisted intervention (MICCAI)
Freund Y, Dasgupta S, Kabra M, Verma N (2007) Learning the structure of manifolds using random projections. In: Advances in neural information processing systems (NIPS)
Gerber S, Tasdizen T, Joshi S, Whitaker R (2009) On the manifold structure of the space of brain images. In: Proc medical image computing and computer assisted intervention (MICCAI)
Geurts P, Ernst D, Wehenkel L (2006) Extremely randomized trees. Mach Learn 36(1)
Gray KR, Aljabar P, Heckeman RA, Hammers A, Rueckert D (2011) Random forest-based manifold learning for classification of imaging data in dementia. In: Proc medical image computing and computer assisted intervention (MICCAI)
Hamm J, Ye DH, Verma R, Davatzikos C (2010) GRAM: a framework for geodesic registration on anatomical manifolds. Med Image Anal 14(5)
Hegde C, Wakin MB, Baraniuk RG (2007) Random projections for manifold learning—proofs and analysis. In: Advances in neural information processing systems (NIPS)
Jolliffe IT (1986) Principal component analysis. Springer, Berlin
Konukoglu E, Glocker B, Zikic D, Criminisi A (2012) Neighborhood approximation forests. In: Proc medical image computing and computer assisted intervention (MICCAI)
Lin Y, Jeon Y (2002) Random forests and adaptive nearest neighbors. J Am Stat Assoc
Marée R, Geurts P, Wehenkel L (2007) Content-based image retrieval by indexing random subwindows with randomized trees. In: Proc Asian conf on computer vision (ACCV). LNCS, vol 4844. Springer, Berlin
Nadler B, Lafon S, Coifman RR, Kevrekidis IG (2005) Diffusion maps, spectral clustering and eigenfunctions of Fokker-Plank operators. In: Advances in neural information processing systems (NIPS)
O’Hara S, Draper BA (2012) Scalable action recognition with a subspace forest. In: Proc IEEE conf computer vision and pattern recognition (CVPR)
Shi T, Horvath S (2006) Unsupervised learning with random forest predictors. J Comput Graph Stat 15
Shi J, Malik J (1997) Normalized cuts and image segmentation. In: Proc IEEE conf computer vision and pattern recognition (CVPR), Washington, DC, USA
Tenenbaum JB, deSilva V, Langford JC (2000) A global geometric framework for nonlinear dimensionality reduction. Science 290(5500)
Xiong C, Johnson D, Xu R, Corso JJ (2012) Random forests for metric learning with implicit pairwise position dependence. In: Proc of ACM SIGKDD intl conf on knowledge discovery and data mining
Zhang Q, Souvenir R, Pless R (2006) On manifold structure of cardiac MRI data: application to segmentation. In: Proc IEEE conf computer vision and pattern recognition (CVPR), Los Alamitos, CA, USA
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer-Verlag London
About this chapter
Cite this chapter
Criminisi, A., Shotton, J. (2013). Manifold Forests. In: Criminisi, A., Shotton, J. (eds) Decision Forests for Computer Vision and Medical Image Analysis. Advances in Computer Vision and Pattern Recognition. Springer, London. https://doi.org/10.1007/978-1-4471-4929-3_7
Download citation
DOI: https://doi.org/10.1007/978-1-4471-4929-3_7
Publisher Name: Springer, London
Print ISBN: 978-1-4471-4928-6
Online ISBN: 978-1-4471-4929-3
eBook Packages: Computer ScienceComputer Science (R0)