Machine Learning — GATE Previous Year Questions

Match the items in Column 1 with the items in Column 2 in the following table:

Euclidean distance based \(k\)-means clustering algorithm was run on a dataset of 100 points with \(k = 3\). If the points \((1,\, 1)^{T}\) and \((-1,\, 1)^{T}\) are both part of cluster 3, then which ONE of the following points is necessarily also part of cluster 3?

Given a dataset with \(K\) binary-valued attributes (where \(K > 2\)) for a two-class classification task, the number of parameters to be estimated for learning a naïve Bayes classifier is

For any binary classification dataset, let \(S_B \in \mathbb{R}^{d \times d}\) and \(S_W \in \mathbb{R}^{d \times d}\) be the between-class and within-class scatter (covariance) matrices, respectively. The Fisher linear discriminant is defined by \(u^{*} \in \mathbb{R}^{d}\), that maximizes

\(J(u) = \dfrac{u^{T} S_B u}{u^{T} S_W u}\)

If \(\lambda = J(u^{*})\), \(S_W\) is non-singular and \(S_B \neq 0\), then \((u^{*}, \lambda)\) must satisfy which ONE of the following equations?

Note: \(\mathbb{R}\) denotes the set of real numbers.

Details of ten international cricket games between two teams “Green” and “Blue” are given in Table C. This table consists of matches played on different pitches, across formats, along with their winners. The attribute Pitch can take one of two values: spin-friendly (represented as \(S\)) or pace-friendly (represented as \(F\)). The attribute Format can take one of two values: one-day match (represented as \(O\)) or test match (represented as \(T\)).

A cricket organization would like to use the information given in Table C to develop a decision-tree model to predict outcomes of future games between these two teams.

To develop such a model, the computed InformationGain(C, Pitch) with respect to the Target is ______ (rounded off to two decimal places).

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 other words, whenever the classifier wrongly predicts a sample \((x_n, y_n)\) from the dataset, \(w_{\mathrm{old}}\) gets updated to \(w_{\mathrm{new}}\), and likewise \(b_{\mathrm{old}}\) gets updated to \(b_{\mathrm{new}}\).

Consider the case \((x_n, +1)\) with \(f(x_n, w_{\mathrm{old}}, b_{\mathrm{old}}) < 0\). Then

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 between \(C_1\) and \(C_2\):

\(\text{DIS-1}(C_1, C_2) = \max_{x \in C_1, \, y \in C_2} D(x, y)\)

\(\text{DIS-2}(C_1, C_2) = \min_{x \in C_1, \, y \in C_2} D(x, y)\)

Which of the following statements is/are correct?

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 ______ (rounded off to three decimal places).

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 \(D = \{x^{(1)}, \dots, x^{(n)}\}\) 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 \(u^{T} u = 1\).

The value of \(\dfrac{1}{n} \sum_{i=1}^{n} \big(u^{T} x^{(i)}\big)^{2}\) is ______ (answer in integer).