Machine Learning - Data Science & Artificial Intelligence Previous Year Questions

Consider the dataset with six datapoints: \( \{(\mathbf{x}_1, y_1), (\mathbf{x}_2, y_2), \ldots, (\mathbf{x}_6, y_6)\} \), where \( \mathbf{x}_1 = \begin{bmatrix} 1 \\ 0 \end{bmatrix} \), \( \mathbf{x}_2 = \begin{bmatrix} 0 \\ 1 \end{bmatrix} \), \( \mathbf{x}_3 = \begin{bmatrix} 0 \\ -1 \end{bmatrix} \), \( \mathbf{x}_4 = \begin{bmatrix} -1 \\ 0 \end{bmatrix} \), \( \mathbf{x}_5 = \begin{bmatrix} 2 \\ 2 \end{bmatrix} \), \( \mathbf{x}_6 = \begin{bmatrix} -2 \\ -2 \end{bmatrix} \), and the labels are given by \( y_1 = y_2 = y_5 = 1 \), and \( y_3 = y_4 = y_6 = -1 \). A hard margin linear support vector machine is trained on the above dataset.
Which ONE of the following sets is a possible set of support vectors?

Consider designing a linear classifier \[ y = \text{sign}(f(x; w, b)), \quad f(x; w, b) = w^\top x + b \] on a dataset \(D = \{(x_1, y_1), (x_2, y_2), \ldots, (x_N, y_N)\}\), \(x_i \in \mathbb{R}^d\), \(y_i \in \{+1, -1\}\), \(i = 1, 2, \ldots, N\). 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: \[ w_{new} = w_{old} + y_n x_n, \quad b_{new} = b_{old} + y_n \] whenever \(\text{sign}(f(x_n; w_{old}, b_{old})) \ne y_n\). In other words, whenever the classifier wrongly predicts a sample \((x_n, y_n)\) from the dataset, \(w_{old}\) gets updated to \(w_{new}\) and likewise \(b_{old}\) gets updated to \(b_{new}\). Consider the case \((x_n, +1), f(x_n; w_{old}, b_{old}) < 0\). Then

Consider that you are training a classifier for a 10-class classification problem. Each input is represented as a 512-dimensional vector. There are 1000 samples, out of which first 100 will be used for testing. Let Leave-One-Out-Cross-Validation (LOOCV) be used for selection of the classifier model before testing. Which of the following options is the correct number of validation splits that will be generated?

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

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 \(C_2\):
\[\mathbf{DIS-1}(C_1, C_2) = \max_{x \in C_1, y \in C_2} D(x, y)\]
\[\mathbf{DIS-2}(C_1, C_2) = \min_{x \in C_1, y \in C_2} D(x, y)\]
Which of the following statements is/are correct?

In the following table, the Task column lists a few tasks related to machine learning. The Algorithm column lists a few algorithms.
Each entry “t” from the Task column is to be matched with an appropriate entry “a” from the Algorithm column such that the task “t” can be solved using the algorithm “a”. Denote such a match as t : a

Task	Algorithm
T1 – Clustering	A1 – Markov Chain Monte Carlo
T2 – Classification	A2 – K-Medoid
T3 – Sampling	A3 – Linear Discriminant Analysis
T4 – Feature Extraction	A4 – Naive Bayes

Which of the following options is/are the correct matching(s)?