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Kernels (Kernalizing (Kernel Function (Similarity (Positive Semidefinite…

- - - - \(\phi\) is in \(D^2\) dimensions
        x and z are in D dimensions
        
        Rewrote expansion -> normal dimensions!
  - - - Similarity between data points
        
        How similar is this one purple dot with another?
      - Weighted vote
      - Positive Semidefinite
        
        Can we use any similarity function? No
        
        K matrix must be positive semidefinite
        
        \(K_{ij} = k(x^{(i)},x^{(j)})\)
        Matrix of all combinations of data points
        
        Similarity function could be "How many words on a page"
        
        The more similar the stronger the weight
      - Dot product \(\approx\) same direction
    - - The larger circles = nonzero weights
        
        On the boundary
        
        Circled triangle is nonzero not on border
- - - - Embed data in higher dimensions
        
        x -> \(\phi (x)\)
    - - Doesn't have to be. When squared it's a squared expansion
        Can be n polynomial expansion
    - - x = D dimensions
      - w = \(D^2\)
        
        Dual form:
      - \(\phi(x) = D^2\)
  - - - \(\phi(x) = D^p\)
      - w = \(D^p\)
        
        Dual form:
      - x = D dimensions
    - - Embed data in higher dimensions
        
        x -> \(\phi (x)\)
    - - Transformation
      - Nonlinear -> Linear
        
        x = degree p polynomial(nonlinear)
        \(\phi(x)\) = linear
    - - MNIST data set dimensions
        784 -> 308504
      - Square and cross match each term
- - - - \(K_{mn} = \phi(x_m)^T\phi(x_n) = \phi(x_n)^T \phi(x_m) = K_{nm}\)
    - - \(a^TKa \geq 0\)
        \((\Phi^Ta)^T(\Phi^Ta) \geq 0\)
        
        Note that \(\Phi\) is the vector form of \(\phi\)