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Kernel nearest means

Kernel nearest means. Usman Roshan. Feature space transformation. Let Φ (x) be a feature space transformation. For example if we are in a two-dimensional vector space and x=(x 1 , x 2 ) then. Computing Euclidean distances in a different feature space.

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Kernel nearest means

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  1. Kernel nearest means Usman Roshan

  2. Feature space transformation • Let Φ(x) be a feature space transformation. • For example if we are in a two-dimensional vector space and x=(x1, x2) then

  3. Computing Euclidean distances in a different feature space • The advantage of kernels is that we can compute Euclidean and other distances in different features spaces without explicitly doing the feature space conversion.

  4. Computing Euclidean distances in a different feature space • First note that the Euclidean distance between two vectors can be written as • In feature space we have where K is the kernel matrix.

  5. Computing distance to mean in feature space • Recall that the mean of a class (say C1) is given by • In feature space the mean Φm would be

  6. Computing distance to mean in feature space

  7. Computing distance to mean in feature space

  8. Computing distance to mean in feature space • Replace K(m,m) and K(m,x) with calculations from previous slides

  9. Kernel nearest means algorithm • Compute kernel • Let xi (i=0..n-1) be the training datapoints and yi (i=0..n’-1) the test. • For each mean mi compute K(mi,mi) • For each datapoint yi in the test set do • For each mean mj do • dj = K(mj,mj) + K(yi,yi) - 2K(mi,yj) • Assign yi to the class with the minimum dj

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