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"## Machine Learning\n",
"### Radial basis functions (RBFs)\n",
"A **radial basis function** (RBF) is a **radial function** which is **positive definite**. It means that the **interpolation** matrix is invertible (nonsingular). A radial function $\\phi$ is only dependent on the distance from a center point $\\boldsymbol{c}$ and not on the direction:\n",
"
$\\large \\phi(\\boldsymbol{x},\\boldsymbol{c})=\\phi(r)$ where $r=||\\boldsymbol{x}-\\boldsymbol{c}||$\n",
"
The radial basis function $\\phi$ is **stricly positive definite** if for any set of distinct numbers $\\boldsymbol{x}_1, \\boldsymbol{x}_2,...,\\boldsymbol{x}_N$; the interpolation matrix defined below is **nonsingular** (all its **eigenvalues** are positive):\n",
"
$\\phi_{ji}=\\phi(||\\boldsymbol{x}_j-\\boldsymbol{x}_i||)$\n",
"
In the following we display three strictly positive definite RBFs.\n",
"