Question 6 For the following sets of two-dimensional points, (1) draw a sketch of how they would be split into clusters by K-means for the given number of clusters and (2) indicate approximately where the resulting centroids would be. Assume that we are using the squared error objective function. If you think that there is more than one possible solution, then please indicate whether each solution is a global or local minimum. Note that the label of each diagram in Figure 1 matches the corresponding part of this question, e.g., Figure 1(a) goes with part (a). Assume that the points in the cluster are of equal density unless mentioned explicitly b a) K-2. How many possible ways are there (in theory) to partition the points into two clusters? What can you say about the positions of the two centroids? (Again, you don’t need to provide exact centroid locations, just a qualitative description.) b) K- 3. Hint: Use the symmetry of the situation c) K 2. Here there are two outlier points with distance d from the center of the circle and d > 2R, where R is the radius of the circle. d) K 2. There are two circles. The bigger circle has only 100 points while the smaller circkle is much denser with 100,000 points. The distance between the two centers is greater than 2R, where R is the radius of the larger circle. Hint: the two centroids are more likely to be initialized with the points of denser circle. e) K 2 Show transcribed image text Question 6 For the following sets of two-dimensional points, (1) draw a sketch of how they would be split into clusters by K-means for the given number of clusters and (2) indicate approximately where the resulting centroids would be. Assume that we are using the squared error objective function. If you think that there is more than one possible solution, then please indicate whether each solution is a global or local minimum. Note that the label of each diagram in Figure 1 matches the corresponding part of this question, e.g., Figure 1(a) goes with part (a). Assume that the points in the cluster are of equal density unless mentioned explicitly b a) K-2. How many possible ways are there (in theory) to partition the points into two clusters? What can you say about the positions of the two centroids? (Again, you don’t need to provide exact centroid locations, just a qualitative description.) b) K- 3. Hint: Use the symmetry of the situation c) K 2. Here there are two outlier points with distance d from the center of the circle and d > 2R, where R is the radius of the circle. d) K 2. There are two circles. The bigger circle has only 100 points while the smaller circkle is much denser with 100,000 points. The distance between the two centers is greater than 2R, where R is the radius of the larger circle. Hint: the two centroids are more likely to be initialized with the points of denser circle. e) K 2

## Expert Answer

Answer to Question 6 For the following sets of two-dimensional points, (1) draw a sketch of how they would be split into clusters …