Chandan Singh | kernels


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An introduction to kernels and recent research.

kernel basics

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

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: $K_Pf(x) = \int K(x, y) f(y) dP(y)$

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 $K_n$)