homework2

# homework2 - Problem Set 2 Due date Wed at 4pm before class...

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Problem Set 2 September 28, 2009 Due date: Wed, October 14 2009 at 4pm; before class. Exercise 1: (20 points) Assume two d-dimensional real vectors x and y . And denote by x i ( y i ) the value in the i -th coordinate of x ( y ). Prove or disprove the following statements: 1. Distance function L 1 ( x, y ) = d X i =1 | x i - y i | is a metric. (5 points) 2. Distance function L 2 ( x, y ) = v u u t d X i =1 ( x i - y i ) 2 is a metric. (5 points) 3. Distance function L 2 2 ( x, y ) = d X i =1 ( x i - y i ) 2 is a metric. (10 points) Exercise 2: (30 points) In class, we have discussed Kleinberg’s impossibility theorem for clustering. Show whether the k -means clustering function satisﬁes (a) richness, (b) scale invariance, and (c) consistency. (10 points for every axiom) Exercise 3: (20 points) The k -means clustering problem takes as input n points X in a d -dimensional space and asks for a partition of the points into k parts C 1 , . . . , C k . Each part C i is represented by a d -dimensional

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homework2 - Problem Set 2 Due date Wed at 4pm before class...

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