1. 1 Kernel-class Jan. 13 2005

2. 2 Recap: Feature Spaces non-linear mapping to F 1. high-D space 2. infinite-D countable space : 3. function space (Hilbert space) example:

3. 3 Recap: Kernel Trick Note: In the dual representation we used the Gram matrix to express the solution. Kernel Trick: Replace : kernel If we use algorithms that only depend on the Gram-matrix, G, then we never have to know (compute) the actual features This is the crucial point of kernel methods

4. 4 Recap: Properties of a Kernel Definition: A finitely positive semi-definite function is a symmetric function of its arguments for which matrices formed by restriction on any finite subset of points is positive semi-definite. Theorem: A function can be written as where is a feature map iff k(x,y) satisfies the semi-definiteness property. Relevance: We can now check if k(x,y) is a proper kernel using only properties of k(x,y) itself, i.e. without the need to know the feature map!

5. 5 Reproducing Kernel Hilbert Spaces The proof of the above theorem proceeds by constructing a very special feature map (note that more feature maps may give rise to a kernel) i.e. we map to a function space. definition function space:reproducing property:

6. 6 Mercer’s Theorem Theorem: X is compact, k(x,y) is symmetric continuous function s.t. is a positive semi-definite operator: i.e. then there exists an orthonormal feature basis of eigen-functions such that: Hence: k(x,y) is a proper kernel. Note: Here we construct feature vectors in l2, where the RKHS construction was in a function space.

7. 7 Modularity Kernel methods consist of two modules: 1) The choice of kernel (this is non-trivial) 2) The algorithm which takes kernels as input Modularity: Any kernel can be used with any kernel-algorithm. some kernels: some kernel algorithms: - support vector machine - Fisher discriminant analysis - kernel regression - kernel PCA - kernel CCA

8. 8 Niceties and Challenges Niceties: Kernel algorithms are typically constrained convex optimization problems  solved with either spectral methods or convex optimization tools. Efficient algorithms do exist in most cases. The similarity to linear methods facilitates analysis. There are strong generalization bounds on test error. Challenges: You need to choose the appropriate kernel Kernel learning is prone to over-fitting All information must go through the kernel-bottleneck.

9. 9 Regularization Demo Trevor Hastie. regularization is very important! regularization parameters typically determined by out of sample. measures (cross-validation, leave-one-out). Example: Gaussian Kernel: if c is very small: G=I (all data are dissimilar): over-fitting if c is very large: G=1 (all data are very similar): under-fitting In RKHS view we compute overlap between 2 Gaussians with width “c”.

10. 10 Learning Kernels All information is tunneled through the Gram-matrix information bottleneck. The real art is to pick an appropriate kernel for the data domain. Warning: Since kernels can overfit, we need to regularize. Solution: We need to learn the kernel. Here is some ways to combine kernels to improve them: k1 k2 cone any positive polynomial parameters can be set by i) cross-validation, ii) Bayesian methods, iii) test-error bound minimization.