RBF kernel
The problem, from first principles
Linear kernel measures similarity with a dot product, which only ever draws straight boundaries. The radial basis function (RBF) kernel, also called the Gaussian kernel, measures similarity by distance instead: two points that are close score near 1, and points far apart score near 0. A classifier built on it can carve out curved, island-shaped regions that no hyperplane could.
Given X with shape (n, d) and Y with shape (m, d), return the (n, m) matrix of RBF similarities with a width parameter gamma.
Constraints
- Return an array of shape
(n, m)where entry[i, j]equalsexp(-gamma * ||X[i] - Y[j]||^2). gammadefaults to1.0.- Every entry must be finite, including the diagonal of
rbf_kernel(X, X), which must be exactly1up to floating point error. - Do not modify
XorY.
Hints
Open one at a time. Each gives away a little more than the last.
Hint 1
Squared distance expands as ||x||^2 + ||y||^2 - 2 x.y. That expansion gives the whole matrix from three pieces, with no double loop.
Hint 2
Floating point rounding can make a squared distance slightly negative for identical points, which would push exp above 1. Clip the squared distances at zero before exponentiating.
Keywords
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