Problems tagged with "kernel ridge regression"

Let \(\nvec{x}{1} = (1, 2, 0)^T\), \(\nvec{x}{2} = (-1, -1, -1)^T\), \(\nvec{x}{3} = (2, 2, 0)^T\), \(\nvec{x}{4} = (0, 2, 0)\).

Suppose a prediction function \(H(\vec x)\) is learned using kernel ridge regression on the above data set using the kernel \(\kappa(\vec x, \vec x') = (1 + \vec x \cdot\vec x')^2\) and regularization parameter \(\lambda = 3\). Suppose that \(\vec\alpha = (1, 0, -1, 2)^T\) is the solution of the dual problem.

Let \(\vec x = (0, 1, 0)^T\) be a new point. What is \(H(\vec x)\)?

Let \(\{\nvec{x}{i}, y_i\}\) be a data set of \(n\) points, with each \(\nvec{x}{i}\in\mathbb R^d\). Recall that the solution to the kernel ridge regression problem is \(\vec\alpha = (K + n \lambda I)^{-1}\vec y\), where \(K\) is the kernel matrix, \(I\) is the identity matrix, \(\lambda > 0\) is a regularization parameter, and \(\vec y = (y_1, \ldots, y_n)^T\).

Suppose kernel ridge regression is performed with a kernel \(\kappa\) that is a kernel for a feature map \(\vec\phi : \mathbb R^d \to\mathbb R^k\).

What is the size of the kernel matrix, \(K\)?

Consider the data set: \(\nvec{x}{1} = (1, 0, 2)^T\)\(\nvec{x}{2} = (-1, 0, -1)^T\)\(\nvec{x}{3} = (1, 2, 1)^T\)\(\nvec{x}{4} = (1, 1, 0)^T\) Suppose a prediction function \(H(\vec x)\) is learned using kernel ridge regression on the above data set using the kernel \(\kappa(\vec x, \vec x') = (1 + \vec x \cdot\vec x')^2\) and regularization parameter \(\lambda = 3\). Suppose that \(\vec\alpha = (-1, -2, 0, 2)^T\) is the solution of the dual problem.