Level 1 — RecognitionUnsupervised Learning

Unsupervised Learning

⏱ 20 minutes30 marksprintable — key stays hidden on paper

Difficulty: Level 1 (Recognition) Time Limit: 20 minutes Total Marks: 30

Section A — Multiple Choice (1 mark each) — 12 marks

Choose the single best answer.

Q1. In the K-Means objective, the algorithm minimizes: (a) the between-cluster variance (b) the sum of squared distances between points and their assigned centroid (WCSS/inertia) (c) the silhouette score (d) the number of clusters $K$

Q2. The "elbow" in the elbow method is identified by plotting: (a) silhouette score vs. number of samples (b) inertia (WCSS) vs. number of clusters $K$ (c) explained variance vs. eigenvalue index (d) density vs. epsilon

Q3. The silhouette score for a single point ranges in: (a) $[0, 1]$ (b) $[-1, 1]$ (c) $[0, \infty)$ (d) $(-\infty, \infty)$

Q4. K-Means++ improves standard K-Means primarily by: (a) guaranteeing the global optimum (b) choosing better initial centroids to spread them out (c) removing the need to specify $K$ (d) using density instead of distance

Q5. In DBSCAN, a point classified as noise is one that: (a) is a core point (b) is density-reachable from a core point (c) is neither a core point nor within $\varepsilon$ of any core point (d) lies exactly on a cluster boundary

Q6. Which linkage method uses the distance between the two closest points of two clusters? (a) complete linkage (b) average linkage (c) single linkage (d) Ward's linkage

Q7. The EM algorithm for Gaussian Mixture Models alternates between: (a) merging and splitting clusters (b) an Expectation step (responsibilities) and a Maximization step (parameter update) (c) forward and backward passes (d) encoding and decoding

Q8. In PCA, the principal components are the: (a) rows of the data matrix (b) eigenvectors of the covariance matrix (c) cluster centroids (d) singular values only

Q9. The proportion of variance explained by a principal component equals: (a) its eigenvalue divided by the sum of all eigenvalues (b) its eigenvector norm (c) the reconstruction error (d) $1/K$

Q10. t-SNE is mainly used for: (a) supervised classification (b) low-dimensional visualization preserving local structure (c) computing exact cluster centroids (d) linear feature scaling

Q11. Compared with t-SNE, UMAP generally: (a) is slower and preserves only local structure (b) is faster and tends to preserve more global structure (c) requires labels to run (d) always produces linear projections

Q12. Agglomerative hierarchical clustering is best described as: (a) top-down: start with one cluster and split (b) bottom-up: start with each point as its own cluster and merge (c) density-based partitioning (d) probabilistic soft assignment

Section B — Matching (1 mark each) — 8 marks

Q13. Match each method (1–8) to its description (A–H). Write pairs like 1→X.

#	Method
1	DBSCAN
2	K-Means
3	GMM (EM)
4	PCA
5	Silhouette score
6	Dendrogram
7	UMAP
8	K-Means++

Letter	Description
A	Tree diagram showing hierarchical merge order
B	Soft probabilistic cluster assignment via Gaussians
C	Smart centroid seeding for spread-out starts
D	Density-based clustering with $\varepsilon$ and minPts
E	Linear projection onto directions of max variance
F	Hard partitioning minimizing within-cluster SSE
G	Cluster-quality metric combining cohesion & separation
H	Nonlinear manifold-based dimensionality reduction

Section C — True/False WITH Justification (2 marks each) — 10 marks

State True or False (1 mark) and give a one-line justification (1 mark).

Q14. "K-Means is guaranteed to find the global minimum of its objective function."

Q15. "DBSCAN requires the number of clusters $K$ to be specified in advance."

Q16. "In PCA the principal components are mutually orthogonal."

Q17. "PCA can be computed via Singular Value Decomposition of the (centered) data matrix, without explicitly forming the covariance matrix."

Q18. "A silhouette score close to $-1$ for a point indicates it is well-clustered."

Answer keyMark scheme & solutions

Section A (1 mark each)

Q1 → (b). K-Means minimizes inertia = $\sum_i \|x_i - \mu_{c_i}\|^2$ , the within-cluster sum of squares.

Q2 → (b). Elbow method plots inertia against $K$ and looks for the bend where marginal decrease drops.

Q3 → (b). Silhouette $s = (b-a)/\max(a,b) \in [-1,1]$ .

Q4 → (b). K-Means++ seeds centroids probabilistically proportional to squared distance, spreading them; it does not guarantee a global optimum.

Q5 → (c). Noise points are neither core nor border (not within $\varepsilon$ of a core point).

Q6 → (c). Single linkage = minimum pairwise distance between clusters.

Q7 → (b). E-step computes responsibilities; M-step updates means, covariances, and mixing weights.

Q8 → (b). Principal components are the eigenvectors of the covariance matrix (equivalently right singular vectors of centered data).

Q9 → (a). Explained variance ratio of component $j$ = $\lambda_j / \sum_k \lambda_k$ .

Q10 → (b). t-SNE is a nonlinear visualization technique emphasizing local neighborhood structure.

Q11 → (b). UMAP is typically faster and retains more global structure than t-SNE.

Q12 → (b). Agglomerative = bottom-up merging.

Section B (1 mark each)

Q13: 1→D, 2→F, 3→B, 4→E, 5→G, 6→A, 7→H, 8→C

Section C (True = 1, justification = 1)

Q14. False. K-Means only converges to a local minimum; result depends on initialization (hence K-Means++/multiple restarts).

Q15. False. DBSCAN does not need $K$ ; it derives clusters from $\varepsilon$ and minPts (density parameters).

Q16. True. Eigenvectors of a symmetric covariance matrix are orthogonal, so principal components are mutually orthogonal.

Q17. True. SVD $X = U\Sigma V^\top$ gives principal directions as columns of $V$ ; variances relate to $\sigma_i^2/(n-1)$ , avoiding forming $X^\top X$ .

Q18. False. A score near $-1$ means the point is likely misassigned (closer to a neighboring cluster); near $+1$ indicates good clustering.

Mark Distribution

Section A: 12 × 1 = 12
Section B: 8 × 1 = 8
Section C: 5 × 2 = 10
Total: 30

[
  {"claim":"Silhouette formula gives value in [-1,1]; test a=1,b=4 -> 0.75","code":"a=1; b=4; s=(b-a)/max(a,b); result=(s==Rational(3,4)) and (-1<=s<=1)"},
  {"claim":"Explained variance ratio: eigenvalues [4,1,0.5,0.5] -> top comp ratio = 4/6","code":"eigs=[4,1,Rational(1,2),Rational(1,2)]; r=eigs[0]/sum(eigs); result=(r==Rational(2,3))"},
  {"claim":"PCA eigenvectors of a symmetric covariance matrix are orthogonal (example)","code":"C=Matrix([[2,0],[0,3]]); v=C.eigenvects(); e1=v[0][2][0]; e2=v[1][2][0]; result=(e1.dot(e2)==0)"},
  {"claim":"Inertia example: points 0,2 with centroid 1 -> SSE=2","code":"pts=[0,2]; c=1; sse=sum((p-c)**2 for p in pts); result=(sse==2)"}
]