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Input Space versus Feature Space in Kernel- Based Methods Scholkopf, Mika, Burges, Knirsch, Muller, Ratsch, Smola presented by: Joe Drish Department of Computer Science and Engineering University of California, San Diego

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Goals 1)Introduce and illustrate the kernel trick 2)Discuss the kernel mapping from input space to feature space F 3)Review kernel algorithms: SVMs and kernel PCA 4)Discuss interpretation of the return from F to after the dot product computation 5)Discuss the form of constructing sparse approximations of feature space expansions 6)Evaluate and discuss the performance of SVMs and PCA Objectives of the paper Applications of kernel methods 1)Handwritten digit recognition 2)Face recognition 3)De-noising: this paper

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Definition A reproducing kernel k is a function k: R. The domain of k consists of the data patterns {x 1, …, x l } is a compact set in which the data lives is typically a subset of R N Computing k is equivalent to mapping data patterns into a higher dimensional space F, and then taking the dot product there. A feature map : R N F is a function that maps the input data patterns into a higher dimensional space F.

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Illustration Using a feature map to map the data from input space into a higher dimensional feature space F: X X X X O O O O Φ(O) Φ(X) F

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Kernel Trick We would like to compute the dot product in the higher dimensional space, or (x) · (y). To do this we only need to compute k(x,y), since k(x,y) = (x) · (y). Note that the feature map is never explicitly computed. We avoid this, and therefore avoid a burdensome computational task.

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Example kernels Gaussian: Polynomial: Sigmoid: Nonlinear separation can be achieved.

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Nonlinear Separation

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Mercer Theory Necessary condition for the kernel-mercer trick: N F is equal to the rank of u i u i T – the outer product is the normalized eigenfunction – analogous to a normalized eigenvector Input Space to Feature Space

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Mercer :: Linear Algebra Linear algebra analogy: Eigenvector problem Eigenfunction problem x and y are vectors u is the normalized eigenvector is the eigenvalue is the normalized eigenfunction Ak(x,y)k(x,y) u, ,

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RKHS, Capacity, Metric Reproducing kernel Hilbert space (RKHS) Hilbert space of functions f on some set X such that all evaluation functions are continuous, and the functions can be reproduced by the kernel Capacity of the kernel map Bound on the how many training examples are required for learning, measured by the VC-dimension h Metric of the kernel map Intrinsic shape of the manifold to which the data is mapped

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Support Vector Machines The decision boundary takes the form: Similar to single layer perceptron Training examples x i with non-zero coefficients i are support vectors

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Kernel Principal Component Analysis KPCA carries out a linear PCA in the feature space F The extracted features take the nonlinear form Theare the components of the k-th eigenvector of the matrix

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KPCA and Dot Products Wish to find eigenvectors V and eigenvalues of the covariance matrix Again, replace (x) · (y). with k(x,y).

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From Feature Space to Input Space Pre-image problem: Here, is not in the image.

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Projection Distance Illustration Approximate the vector F:

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Minimizing Projection Distance Maximize: z is an approximate pre-image for if: For kernels where k(z,z) = 1 (Gaussian), this reduces to:

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Fixed-point iteration Requiring no step-size, we can iterate: So assuming a Gaussian kernel: i are the eigenvectors of the centered Gram matrix x i are the input space is the width

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Kernel PCA Toy Example Generated an artificial data set from three point sources, 100 point each.

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De-noising by Reconstruction, Part One Reconstruction from projections onto the eigenvectors from previous example Generated 20 new points from each Gaussian Represented by their first n = 1, 2, …, 8 nonlinear principal components

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De-noising by Reconstruction, Part Two Original points are moving in the direction of de-noising

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De-noising in 2-dimensions A half circle and a square in the plane De-noised versions are the solid lines

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De-noising USPS data patterns Patterns 7291 train 2007 test Size: 16 x 16 Linear PCA Kernel PCA

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Questions

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