Machine Learning Techniques Quiz 1: 26 October 2025 (September 2025 term)

Which among the following is true with respect to the Lloyds’ algorithm?

Consider the dataset consisting of the following points:

$(0, 0), \; (0, 2), \; (2, 0), \; (2, 2).$

We want to run the $K$-means++ algorithm with $K = 2$. Suppose the first cluster center chosen is $(0, 0)$. Which of the following statements are correct?

Hint: The score of a datapoint $x$ is given by $S(x) = \min_{j=1,2,\ldots,l-1} (||x - \mu_j^0||^2)$, where $\mu_j^0$ is the initial cluster center.

Suppose that 100 items are sampled from a manufacturing process, where three items are found to be defective. Consider $\text{Beta}(a, b)$ as a prior distribution for the proportion $\theta$ of defective items with the following parameters:

• $a = 1, b = 1$
• $a = 0.5, b = 0.5$

If posterior mean is used as the estimate for $\theta$, then which of the following is correct?

Hint: Expectation of a $\text{Beta}(a, b)$ is given by $\frac{a}{(a + b)}$.

Which of the following are valid kernel functions? Select all that apply.

Consider a mean-centered dataset consisting of 150 patients from the health-care domain with 6 features. The dataset is represented as a $150 \times 6$ matrix $X$. PCA is performed on this dataset, and it is observed that the reconstruction error (residue) vanishes completely after $k$ components. A domain expert provides the following insights about the columns of $X$, denoted by $c_1, c_2, \ldots, c_6$:

(a) The set $\{c_1, c_2, c_3\}$ is linearly independent.

(b) The following relationships hold:

• $c_4 = 2c_1 - c_2$
• $c_5 = c_2 + c_3$
• $c_6 = 4c_1 - 3c_3$

What is the value of $k$?

Consider a dataset $X \in \mathbb{R}^{d \times n}$ with sample size $n = 100$ (i.e., $X$ has $d$ features and 100 examples), where $d \gg 100$. Suppose the first five eigenvalues of the matrix $X^TX$ are:

$10,\ 5,\ 3,\ 2,\ 1,$

and all remaining eigenvalues are zero.

What is the proportion of variance explained along the first principal component? Enter your answer correct to two decimal places.

Based on the above data, answer the given subquestions.

What is the covariance matrix of the dataset?

Based on the above data, answer the given subquestions.

If standard PCA is performed on this dataset, what is the variance along the first principal component? Enter the answer correct to one decimal place.

Based on the above data, answer the given subquestions.

If standard PCA is performed on this dataset, what is the variance along the second principal component? Enter the answer correct to one decimal place.

Based on the above data, answer the given subquestions.

Based on the above data, answer the given subquestions.

Suppose at the $t$-th iteration of the EM algorithm, the parameters are:

$\pi_1 = 0.3, \quad \pi_2 = 0.7, \quad \mu_1 = 2, \quad \mu_2 = 3, \quad \sigma_1^2 = 1, \quad \sigma_2^2 = 1.$

The densities of the data points given each mixture component are provided below:

What is the value of $\lambda_1^1$? Enter your answer correct to two decimal places.

Based on the above data, answer the given subquestions.

Perform K-means clustering with K = 2 and initial cluster centers at (−2, 3) and (2,−1). What are the final means of clusters after convergence?

Based on the above data, answer the given subquestions.

Suppose we introduce a new data point (0, 0) to the existing dataset and perform K means with the same initial cluster centers. How many iterations would it require now for convergence?