Question 1

Figure from the original question paper Table of training data x_i, y_i for linear regression, with the question text Consider the following dataset for a kernel regression problem with a polynomial kernel of degree two along with the coefficient vector $\boldsymbol{\alpha}$: $$\mathbf{X} = \begin{bmatrix} 1 & 0 & -1 & 0 \\ 0 & 1 & 0 & -1 \end{bmatrix}, \mathbf{y} = \begin{bmatrix} 1 \\ -1 \\ 2 \\ -2 \end{bmatrix}, \boldsymbol{\alpha} = \begin{bmatrix} \alpha_1 \\ -0.625 \\ \alpha_3 \\ \alpha_4 \end{bmatrix}$$ If the prediction for the data-point $\begin{bmatrix} 1 \\ 1 \end{bmatrix}$ is 0, find the value of $\alpha_1$. \_\_\_\_\_\_\_\_\_\_