Question

In this problem, we will work through K-means for a single-dimensional data. (a) Consider the case where K = 3 and we have 4 data points x1 = 1, x2 = 2, x3 = 5, x4 = 7. What is the optimal clustering for this data? What is the corresponding value of the K-means objective? (b) One might be tempted to think that Lloyd’s algorithm is guaranteed to converge to the global minimum when dimension d = 1. Show that there exists a suboptimal cluster assignment (i.e., initialization) for the data in the above part that Lloyd’s algorithm will not be able to improve (to get full credit, you need to show the assignment, show why it is suboptimal and explain why it will not be improved).

          In this problem, we will work through K-means for a single-dimensional data.
(a) Consider the case where K = 3 and we have 4 data points x1 = 1, x2 = 2, x3 = 5, x4 = 7.
What is the optimal clustering for this data? What is the corresponding value of the K-means objective?
(b) One might be tempted to think that Lloyd’s algorithm is guaranteed to converge to the global minimum when dimension d = 1. Show that there exists a suboptimal cluster assignment (i.e., initialization) for the data in the above part that Lloyd’s algorithm will not be able to improve (to get full credit, you need to show the assignment, show why it is suboptimal and explain why it will not be improved).

Added by Paul V.

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