MMiDS 1.3: Self-Assessment Quiz

Which of these is NOT a property of a valid partition \( C_1, \ldots, C_k \) in the context of \( k \)-means?

• a) The subsets are pairwise disjoint
• b) The subsets cover all data points
• c) Each subset is non-empty
• d) Each subset contains an equal number of points

In the \( k \)-means objective function, what does the variable \( \boldsymbol{\mu}_i \) represent?

• a) The centroid of cluster \( i \)
• b) The number of points in cluster \( i \)
• c) The distance between clusters \( i \) and \( j \)
• d) The assignment of point \( j \) to a cluster

The \( k \)-means objective function is a measure of what?

• a) The total number of clusters.
• b) The average distance between data points.
• c) The sum of squared distances between each data point and its assigned cluster center.
• d) The maximum distance between any two cluster centers.

How does the \( k \)-means algorithm update the partition in each iteration?

• a) By randomly reassigning points to clusters
• b) By assigning each point to the cluster whose current centroid is closest
• c) By moving the cluster centroids to random new locations
• d) By merging the two closest clusters

Which of the following statements about the \( k \)-means algorithm is TRUE?

• a) It is guaranteed to find the global minimum of the k-means objective function.
• b) It is a heuristic that may not find the global minimum.
• c) It always converges to the same solution regardless of initialization.
• d) It is primarily used for hierarchical clustering.

What is a key property of the sequence of objective function values produced by the \( k \)-means algorithm?

• a) It is strictly decreasing
• b) It is non-increasing
• c) It is strictly increasing
• d) It alternates between two values

Which mathematical property is essential in the equivalence of minimizing \( \|\mathbf{x}_j - \boldsymbol{\mu}_i\|^2 \) and \( \|\mathbf{x}_j - \boldsymbol{\mu}_i\| \)?

• a) Non-decreasing nature of square and square root functions
• b) Decreasing nature of exponential functions
• c) Linearity of the Euclidean distance
• d) Independence of data points

What is the interpretation of the matrix \( Z \) in the matrix formulation of \( k \)-means?

• a) It represents the cluster centers.
• b) It represents the distances between data points.
• c) It encodes the cluster assignments of each data point.
• d) It represents the covariance matrix of the data.

What property must a matrix \( Z \) representing cluster assignments satisfy?

• a) Each row must sum to 1
• b) Each column must sum to 1
• c) Each row must have exactly one 1 and the rest 0s
• d) Each column must have exactly one 1 and the rest 0s

Why is standardization often applied to data before running k-means clustering?

• a) To ensure all features have equal weight in the clustering process.
• b) To reduce the computational complexity of the algorithm.
• c) To make the algorithm less sensitive to outliers.
• d) To transform the data into a Gaussian distribution.

What is the significance of the Frobenius norm in the context of \( k \)-means clustering?

• a) It measures the intra-cluster variance
• b) It quantifies the distance between the data matrix and the matrix formed by cluster centroids
• c) It is used to normalize data points before clustering
• d) It measures the likelihood of data points belonging to a cluster