5.1. kernels

An introduction to kernels and recent research.

5.1.1. kernel basics

5.1.1.1. ch 4 from support vector machines book

  • 4.1 - what is a valid kernel

    • in general, most dot-product like things constitute valid kernels

    • a function is a kernel iff it is a symmetric, positive definite function

      • this refers to the \(n\) x \(n\) matrix with entries \(f(x_{row}-x_{col})\) being a psd matrix

    • a given kernel can have many feature spaces (can construct different feature spaces that yield the same inner products)

  • 4.2 reproducing kernel hilbert space (RKHS) of a kernel

    • hilbert space - abstract vector space with (1) an inner product and (2) is complete (i.e. enough limits so calculus works)

    • the RKHS is the smallest feature space of a kernel, and can serve as a canonical feature space

      • RKHS - a \(\mathbb K\)-hilbert space that consists of functions mapping from X to \(\mathbb K\)

      • every RKHS has a unique reproducing kernel

      • every kernel has a unique RKHS

    • sums/products of kernels also work

5.1.2. kernel papers

  • data spectroscopy paper (shi et al. 2009)

    • kernel matrix \(K_n\)

    • Laplacian matrix \(L_n = D_n - K_n\)

      • \(D_n\) is diagonal matrix, with entries = column sums

    • block-diagonal kernel matrix would imply a cluster

      • eigenvalues of these kernel matrices can identify clusters (and are invariant to permutation)

      • want to look for data points corresponding to same/similar eigenvectors

    • hard to know what kernel to use, how many eigenvectors / groups look at

    • here, look at population point of view - realted dependence of spectrum of \(K_n\) on the data density function: \(K_Pf(x) = \int K(x, y) f(y) dP(y)\)

5.1.3. spectral clustering

  • interested in top eigenvectors of \(K_n\) and bottom eigenvectors of \(L_n\)

  • scott and longuet-higgins - embed data in space of top eigenvectors, normalize in that space, and investigate block structure

  • perona and freeman - 2 clusters by thresholding top eigenvector

  • shi & malik - normalized cut: threshold second smallest generalize eigenvector of \(L_n\)

  • similarly we have kernel PCA, spectral dimensionality reduction, and SVMs (which can be viewed as fitting a linear classifier in the eigenspace of \(K_n\))