Year-wise GATE papers
5 questions and papers
Let \(D = \{x^{(1)}, \dots, x^{(n)}\}\) be a dataset of \(n\) observations where each \(x^{(i)} \in \mathbb{R}^{100}\). It is given that \(\sum_{i=1}^{n} x^{(i)} = 0\). The covariance matrix computed from \(D\) has eigenvalues \(\lambda_i = 100^{\,2-i}\), \(1 \le i \le 100\). Let \(u \in \mathbb{R}^{100}\) be the direction of maximum variance with…
Consider designing a linear binary classifier \(f(x) = \mathrm{sign}(w^{T} x + b)\), \(x \in \mathbb{R}^{2}\), on the following training data:Class-1: \((2,\, 0)^{T}, (0,\, 2)^{T}, (2,\, 2)^{T}\)Class-2: \((0,\, 0)^{T}\)Hard-margin support vector machine (SVM) formulation is solved to obtain \(w\) and \(b\). Which of the following options is/are correct?
Let \(C_1\) and \(C_2\) be two sets of objects. Let \(D(x, y)\) be a measure of dissimilarity between two objects \(x\) and \(y\). Consider the following definitions of dissimilarity…
Given data \(\{(-1, 1), (2, -5), (3, 5)\}\) of the form \((x, y)\), we fit a model \(y = wx\) using linear least-squares regression. The optimal value of \(w\) is ______…
Consider designing a linear classifier\(y = \mathrm{sign}\big(f(x, w, b)\big), \qquad f(x, w, b) = w^{T} x + b\)on a dataset \(D = \{(x_1, y_1), (x_2, y_2), \dots, (x_N, y_N)\}\), with \(x_i \in \mathbb{R}^{d}\), \(y_i \in \{+1, -1\}\), \(i = 1, 2, \dots, N\). Recall that the sign function outputs \(+1\) if the argument is positive, and \(-1\) if the argument is non-positive. The parameters \(w\) and \(b\) are updated as per the following training algorithm:\(w_{\mathrm{new}} = w_{\mathrm{old}} + y_n x_n, \qquad b_{\mathrm{new}} = b_{\mathrm{old}} + y_n\)whenever \(\mathrm{sign}\big(f(x_n, w_{\mathrm{old}}, b_{\mathrm{old}})\big) \neq y_n\). In…
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