Question: Given m data points x', i = 1, ...,m, K-means clustering algorithm groups them into k clusters by mini- mizing the distortion function over {rij,

Given m data points x', i = 1, ...,m, K-means clustering algorithm groups them into k clusters by mini- mizing the distortion function over {rij, u'} m k J = EEmix - we'll?, (1) i=1j=1 where rij = 1 if x' belongs to the j-th cluster and rij = 0 otherwise. 1. (3 points) Derive mathematically that using the squared Euclidean distance |/x - w |2 as the dissim- ilarity function, the centroid that minimizes the distortion function J for given assignments r are given by Erij That is, p' is the center of j-th cluster. Hint: You may start by taking the partial derivative of J with respect to p, with rij fixed. 2. (2 points) Derive mathematically what should be the assignment variables r' be to minimize the distortion function J, when the centroids ' are fixed

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