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Nonlinear Dimension Reduction Presenter: Xingwei Yang The powerpoint is organized from: 1.Ronald R. Coifman et al. (Yale University) 2. Jieping Ye, (Arizona State University)

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Motivation Linear projections will not detect the pattern.

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Nonlinear PCA using Kernels Traditional PCA applies linear transformation May not be effective for nonlinear data Solution: apply nonlinear transformation to potentially very high- dimensional space. Computational efficiency: apply the kernel trick. Require PCA can be rewritten in terms of dot product. More on kernels later

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Nonlinear PCA using Kernels Rewrite PCA in terms of dot product The covariance matrix S can be written as Let v be The eigenvector of S corresponding to nonzero eigenvalue Eigenvectors of S lie in the space spanned by all data points.

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Nonlinear PCA using Kernels The covariance matrix can be written in matrix form: Any benefits?

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Nonlinear PCA using Kernels Next consider the feature space: The (i,j)-th entry of is Apply the kernel trick: K is called the kernel matrix.

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Nonlinear PCA using Kernels Projection of a test point x onto v: Explicit mapping is not required here.

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Diffusion distance and Diffusion map A symmetric matrix M s can be derived from M as M and M s has same N eigenvalues, Under random walk representation of the graph M : left eigenvector of M : right eigenvector of M : time step 8/14

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If one starts random walk from location x i, the probability of landing in location y after r time steps is given by For large , all points in the graph are connected (M i,j >0) and the eigenvalues of M Diffusion distance and Diffusion map has the dual representation (time step and kernel width). where e i is a row vector with all zeros except that i th position = 1.

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Diffusion distance and Diffusion map One can show that regardless of starting point x i Left eigenvector of M with eigenvalue 0 =1 with Eigenvector 0 ( x ) has the dual representation : 1. Stationary probability distribution on the curve, i.e., the probability of landing at location x after taking infinite steps of random walk (independent of the start location). 2. It is the density estimate at location x.

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Diffusion distance For any finite time r, k and k are the right and left eigenvectors of graph Laplacian M. is the k th eigenvalue of M r (arranged in descending order). Given the definition of random walk, we denote Diffusion distance as a distance measure at time t between two pmfs as with empirical choice w(y)=1/ 0 (y).

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Diffusion Map Diffusion distance : Diffusion map : Mapping between original space and first k eigenvectors as Relationship : This relationship justifies using Euclidean distance in diffusion map space for spectral clustering. Since, it is justified to stop at appropriate k with a negligible error of order O( k+1 / k ) t ).

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Example: Hourglass

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Example: Image imbedding

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Example: Lip image

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Shape description

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Dimension Reduction of Shape space

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References Unsupervised Learning of Shape Manifolds (BMVC 2007) Diffusion Maps(Appl. Comput. Harmon. Anal. 21 (2006)) Geometric diffusions for the analysis of data from sensor networks (Current Opinion in Neurobiology 2005)

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