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📗 [4 points] Given two instances \(x_{1}\) = and \(x_{2}\) = , suppose the feature map for a kernel SVM (Support Vector Machine) is \(\varphi\left(x\right)\) = , what is the kernel (Gram) matrix?
📗 Answer (matrix with multiple lines, each line is a comma separated vector): .
📗 [4 points] Consider a kernel \(K\left(x, y\right)\) = + , where both \(x\) and \(y\) are positive real numbers. What is the feature vector \(\varphi\left(x\right)\) induced by this kernel evaluated at \(x\) = ?
📗 Answer (comma separated vector): .
📗 [4 points] Consider a kernel \(K\left(x, y\right)\) = + + , where both \(x\) and \(y\) are positive real numbers. What is the feature vector \(\varphi\left(x\right)\) induced by this kernel evaluated at \(x\) = ?
📗 Answer (comma separated vector): .
📗 [3 points] Given there are data points, each data point has features, the feature map creates new features (to replace the original features). What is the size of the kernel matrix when training a kernel SVM (Support Vector Machine)? For example, if the matrix is \(2 \times 2\), enter the number \(4\).
📗 Answer: .
📗 [4 points] Given the two training points and and their labels \(0\) and \(1\). What is the kernel (Gram) matrix if the RBF (radial basis function) Gaussian kernel with \(\sigma\) = is used? Use the formula \(K_{i i'} = e^{- \dfrac{1}{2 \sigma^{2}} \left(x_{i} - x_{i'}\right)^\top \left(x_{i} - x_{i'}\right)}\).
📗 Answer (matrix with multiple lines, each line is a comma separated vector): .
📗 [2 points] Let \(w\) = and \(b\) = . For the point \(x\) = , \(y\) = , what is the smallest slack value \(\xi\) for it to satisfy the margin constraint?
📗 Answer: .
📗 [4 points] If \(K\left(x, x'\right)\) is a kernel with induced feature representation \(\varphi\left(x_{0}\right)\) = , and \(G\left(x, x'\right)\) is another kernel with induced feature representation \(\theta\left(x_{0}\right)\) = , then it is known that \(H\left(x, x'\right) = a K\left(x, x'\right) + b G\left(x, x'\right)\), \(a\) = , \(b\) = is also a kernel. What is the induced feature representation of \(H\) for this \(x_{0}\)?
📗 Answer (comma separated vector): .
📗 [3 points] Recall a SVM (Support Vector Machine) with slack variables has the objective function \(\dfrac{\lambda}{2} w^\top w + \dfrac{1}{n} \displaystyle\sum_{i=1}^{n} \xi_{i}\), which is equivalent to \(\dfrac{1}{2} w^\top w + C \displaystyle\sum_{i=1}^{n} \xi_{i}\). What is the optimal \(w\) when the trade-off parameter \(C\) is 0? The training data contains only points with label 0 and with label 1. Only enter the weights, no bias.
📗 Answer (comma separated vector): .
📗 [3 points] Recall a linear SVM (Support Vector Machine) with slack variables has the objective function \(\dfrac{1}{2} w^\top w + C \displaystyle\sum_{i=1}^{n} \varepsilon_{i}\). What is the optimal \(w\) when the trade-off parameter \(C\) is 0? The training data contains only points with label 0 and with label 1. Only enter the weights, no bias.
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