cs2007_2026T2_Q1_NA.pdf

Machine Learning Techniques(MLT) · Quiz 1 · May 2026

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Question 2 MCQ · 4.0 marks

Consider a mean-centred dataset [[IMAGE:41da6c7104319065_2_2]] consisting of **5000** points in [[IMAGE:41da6c7104319065_2_3]] with features [[IMAGE:41da6c7104319065_2_4]] .Every data point satisfies [[IMAGE:41da6c7104319065_2_5]] The independent base feature [[IMAGE:41da6c7104319065_2_6]] has variance [[IMAGE:41da6c7104319065_2_7]] . The remaining features [[IMAGE:41da6c7104319065_2_8]] are mutually uncorrelated, uncorrelated with [[IMAGE:41da6c7104319065_2_9]] , and each has variance [[IMAGE:41da6c7104319065_2_10]] .Standard PCA is performed on this dataset. Let [[IMAGE:41da6c7104319065_2_11]] denote the eigenvalues of the covariance matrix in descending order. Which of the following statements is strictly true?

Exactly **3** eigenvalues are zero: [[IMAGE:41da6c7104319065_2_12]] , and [[IMAGE:41da6c7104319065_2_13]]
Exactly **4** eigenvalues are zero( [[IMAGE:41da6c7104319065_2_14]] ), and [[IMAGE:41da6c7104319065_2_15]]
Exactly **4** eigenvalues are zero( [[IMAGE:41da6c7104319065_2_16]] ), and [[IMAGE:41da6c7104319065_2_17]]
Exactly **2** eigenvalues are zero( [[IMAGE:41da6c7104319065_2_18]] ), and [[IMAGE:41da6c7104319065_2_19]] lies somewhere in the [[IMAGE:41da6c7104319065_3_20]] - [[IMAGE:41da6c7104319065_3_21]] - [[IMAGE:41da6c7104319065_3_22]] subspace.

A published solution is not available for this question yet.

Question 3 MCQ · 4.0 marks

Let [[IMAGE:41da6c7104319065_3_23]] .Define [[IMAGE:41da6c7104319065_3_24]] Which of the following is the correct expression for [[IMAGE:41da6c7104319065_3_25]] ?

[[IMAGE:41da6c7104319065_3_26]]
[[IMAGE:41da6c7104319065_3_27]]
[[IMAGE:41da6c7104319065_3_28]]
[[IMAGE:41da6c7104319065_3_29]]

A published solution is not available for this question yet.

Question 4 MCQ · 4.0 marks

Consider the one-dimensional dataset [[IMAGE:41da6c7104319065_3_30]] Suppose [[IMAGE:41da6c7104319065_3_31]] -means is run with [[IMAGE:41da6c7104319065_3_32]] clusters. At iteration [[IMAGE:41da6c7104319065_3_33]] , the centroids are [[IMAGE:41da6c7104319065_3_34]] The assignment step is performed using these centroids to produce new assignments [[IMAGE:41da6c7104319065_3_35]] for each point. Compute [[IMAGE:41da6c7104319065_3_36]]

A published solution is not available for this question yet.

Question 5 MCQ · 3.0 marks

Which of the following statements is [[IMAGE:41da6c7104319065_4_37]] true for the [[IMAGE:41da6c7104319065_4_38]] -means algorithm with [[IMAGE:41da6c7104319065_4_39]] cluster centres, between consecutive iterations [[IMAGE:41da6c7104319065_4_40]] and [[IMAGE:41da6c7104319065_4_41]] ?

If [[IMAGE:41da6c7104319065_4_42]] ,then at least one data point will change its cluster assignment from iteration [[IMAGE:41da6c7104319065_4_43]] to [[IMAGE:41da6c7104319065_4_44]] .
If [[IMAGE:41da6c7104319065_4_45]] , then no data point will change its cluster assignment from iteration [[IMAGE:41da6c7104319065_4_46]] to [[IMAGE:41da6c7104319065_4_47]] .
If a data point changes its cluster assignment from cluster [[IMAGE:41da6c7104319065_4_48]] to cluster [[IMAGE:41da6c7104319065_4_49]] , it is guaranteed that the centroid of cluster [[IMAGE:41da6c7104319065_4_50]] strictly changes, i.e., [[IMAGE:41da6c7104319065_4_51]] .
It is not possible for the [[IMAGE:41da6c7104319065_4_52]] -means algorithm to revisit a configuration, where a configuration corresponds to the [[IMAGE:41da6c7104319065_4_53]] -way partition generated by clustering at the end of each iteration.

A published solution is not available for this question yet.

Question 6 MCQ · 3.0 marks

A dataset containing 1,000 points is clustered using the [[IMAGE:41da6c7104319065_4_54]] -means algorithm. Which of the following values of [[IMAGE:41da6c7104319065_5_55]] is most likely to yield the [[IMAGE:41da6c7104319065_5_56]] value of objective function of [[IMAGE:41da6c7104319065_5_57]] -means?

[[IMAGE:41da6c7104319065_5_58]]
[[IMAGE:41da6c7104319065_5_59]]
[[IMAGE:41da6c7104319065_5_60]]
[[IMAGE:41da6c7104319065_5_61]]

A published solution is not available for this question yet.

Question 7 MCQ · 3.0 marks

Consider the following kernel function defined for scalar inputs [[IMAGE:41da6c7104319065_5_62]] : [[IMAGE:41da6c7104319065_5_63]] Which of the following feature maps [[IMAGE:41da6c7104319065_5_64]] satisfies [[IMAGE:41da6c7104319065_5_65]] for all [[IMAGE:41da6c7104319065_5_66]] ?

[[IMAGE:41da6c7104319065_5_67]]
[[IMAGE:41da6c7104319065_5_68]]
[[IMAGE:41da6c7104319065_5_69]]
[[IMAGE:41da6c7104319065_5_70]]

A published solution is not available for this question yet.

Question 8 MSQ · 4.0 marks

Let [[IMAGE:41da6c7104319065_6_71]] be a convex function.Which of the following choices of [[IMAGE:41da6c7104319065_6_72]] [[IMAGE:41da6c7104319065_6_73]] that [[IMAGE:41da6c7104319065_6_74]] holds for [[IMAGE:41da6c7104319065_6_75]] convex function [[IMAGE:41da6c7104319065_6_76]] and [[IMAGE:41da6c7104319065_6_77]] [[IMAGE:41da6c7104319065_6_78]] ?

[[IMAGE:41da6c7104319065_6_79]]
[[IMAGE:41da6c7104319065_6_80]]
[[IMAGE:41da6c7104319065_6_81]]
[[IMAGE:41da6c7104319065_6_82]]
[[IMAGE:41da6c7104319065_6_83]]

A published solution is not available for this question yet.

Question 9 MSQ · 4.0 marks

Assume that [[IMAGE:41da6c7104319065_6_84]] are valid (positive semi-definite) kernels. Which of the following are [[IMAGE:41da6c7104319065_6_85]] valid kernels?

[[IMAGE:41da6c7104319065_6_86]]
[[IMAGE:41da6c7104319065_6_87]]
[[IMAGE:41da6c7104319065_6_88]]
[[IMAGE:41da6c7104319065_6_89]]

A published solution is not available for this question yet.

Question 10 MSQ · 4.0 marks

Consider a GMM with [[IMAGE:41da6c7104319065_7_90]] components.During the E-step for a single data point [[IMAGE:41da6c7104319065_7_91]] , you are given only the following two quantities for the first component: [[IMAGE:41da6c7104319065_7_92]] where [[IMAGE:41da6c7104319065_7_93]] is the prior mixing weight and [[IMAGE:41da6c7104319065_7_94]] is the posterior responsibility. All other component parameters and densities are unknown. Based strictly on these two values and the mathematical definition of the E-step responsibility, [[IMAGE:41da6c7104319065_7_95]] which of the following quantities can be uniquely determined as exact numerical values?

The marginal probability density of the data point, [[IMAGE:41da6c7104319065_7_96]] .
The ratio of the conditional density to the marginal density for component [[IMAGE:41da6c7104319065_7_97]] : [[IMAGE:41da6c7104319065_7_98]] .
The ratio of the joint probability of component [[IMAGE:41da6c7104319065_7_99]] to the sum of joint [[IMAGE:41da6c7104319065_7_100]] probabilities of all other components: .
The sum of posterior responsibilities for all components other than [[IMAGE:41da6c7104319065_7_102]] component [[IMAGE:41da6c7104319065_7_101]] :

A published solution is not available for this question yet.

Question 11 NAT · 4.0 marks

Consider a two-component Gaussian Mixture Model (GMM) with mixing weights [[IMAGE:41da6c7104319065_8_103]] and [[IMAGE:41da6c7104319065_8_104]] .For a particular observation [[IMAGE:41da6c7104319065_8_105]] , the component conditional densities satisfy [[IMAGE:41da6c7104319065_8_106]] During the E-step of the EM algorithm, the posterior responsibility of component [[IMAGE:41da6c7104319065_8_107]] is computed as [[IMAGE:41da6c7104319065_8_108]] .Determine the value of [[IMAGE:41da6c7104319065_8_109]] , correct to two decimal places.

A published solution is not available for this question yet.

Question 12 NAT · 3.0 marks

Let [[IMAGE:41da6c7104319065_8_110]] be a dataset of [[IMAGE:41da6c7104319065_8_111]] observations where each [[IMAGE:41da6c7104319065_8_112]] .It is given that [[IMAGE:41da6c7104319065_8_113]] The covariance matrix computed from [[IMAGE:41da6c7104319065_8_114]] has eigenvalues [[IMAGE:41da6c7104319065_8_115]] Let [[IMAGE:41da6c7104319065_8_116]] be the direction of maximum variance with [[IMAGE:41da6c7104319065_8_117]] . Find the value of [[IMAGE:41da6c7104319065_8_118]] .

A published solution is not available for this question yet.