Kernels & the kernel trick

What the problems look like

A few statements from this skill. Answers, hints and worked solutions live in the practice session.

1. Warm-up

   With the homogeneous quadratic kernel $k(\mathbf{u}, \mathbf{v}) = (\mathbf{u}^{\top}\mathbf{v})^{2}$, the Gram matrix of the points $\mathbf{x}_1 = (1, 3)$ and $\mathbf{x}_2 = (3, 1)$ has entries $G_{ij} = k(\mathbf{x}_i, \mathbf{x}_j)$. Compute the off-diagonal entry $G_{12}$.

2. Warm-up

   With the homogeneous quadratic kernel $k(\mathbf{u}, \mathbf{v}) = (\mathbf{u}^{\top}\mathbf{v})^{2}$, the Gram matrix of the points $\mathbf{x}_1 = (2, 3)$ and $\mathbf{x}_2 = (2, 3)$ has entries $G_{ij} = k(\mathbf{x}_i, \mathbf{x}_j)$. Compute the off-diagonal entry $G_{12}$.

3. Warm-up

   With the homogeneous quadratic kernel $k(\mathbf{u}, \mathbf{v}) = (\mathbf{u}^{\top}\mathbf{v})^{2}$, the Gram matrix of the points $\mathbf{x}_1 = (3, 1)$ and $\mathbf{x}_2 = (3, 1)$ has entries $G_{ij} = k(\mathbf{x}_i, \mathbf{x}_j)$. Compute the off-diagonal entry $G_{12}$.

Problems shown under their original terms: Generated (template, solver-computed).