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  "body": "\n## Summary\n\nKernel approximation reconstructs a function from its values at finitely many points $x_j$.\nIt uses a combination $\\sum_j c_j K(\\cdot, x_j)$ of copies of a positive definite kernel $K$.\nThe natural setting is the [[reproducing kernel Hilbert space|4CK2GSRAGD]] $H_K$ of the kernel.\nThere the kernel interpolant is the element of smallest norm that matches the data.\nIts pointwise error is bounded by a power function times the norm of the target.\nIt covers radial basis function interpolation and works in many variables and on scattered points.\n\n## Explanations\n\n### L1 human\n\nPlace a bump, given by the kernel, at each data point, and choose the heights of the bumps so that their sum passes through the data.\nFor the Gaussian kernel $K(x, y) = e^{-\\abs{x - y}^2}$ and one data point $x_1 = 0$ with value $1$, the interpolant is just $e^{-x^2}$.\nWith more points one solves a linear system with the matrix of kernel values $K(x_i, x_j)$.\nThe reproducing kernel Hilbert space explains why this is a good choice.\nAmong all functions of the space that fit the data, this sum is the smoothest in the sense of the norm.\nAdding a penalty to the fit gives kernel ridge regression and other methods of machine learning.\n",
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