K means ++ and K means ||

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1.K means ++ and K means Parallel Jun Wang

2.Review of K means Simple and fast Choose k centers randomly Class points to its nearest center Update centers

3.K means ++ Actually you are using it Spend some time on choosing k centers(seeding) Save time on clustering

4.K means ++ algorithm Seeding Choose a center from X randomly For k-1 times Sample one center each time from X with probability p Update center matrix Clustering

5.d i 2 =min( euclidean distance b/t Xi to each Ci )

6.How to choose K centers

7.Choose a point from X randomly

8.Calculate all d i 2

9.Calculate Pi D=d 1 2 +d 2 2 +d 3 2 + …+ d n 2 P i =d i 2 / D ∑P i =1 Points further away from red point have better chance to be chosen

10.Pick up point with probability p

11.Keep doing the following: Update center matrix Calculate d i 2 Calculate pi Until k centers are found

12.K means || algorithm Seeding Choose a small subset C from X Assign weight to points in C Cluster C and get k centers Clustering

13.Choose subset C from X Let D=Sum of square distance=d 1 2 +d 2 2 +d 3 2 + …+ d n 2 Let L be f(k) like 0.2k or 1.5k for ln(D) times Pick up each point in X using Bernoulli distribution P(chosen)= L *d i 2 /D Update the C

14.How many data in C?

15.How many data in C? Ln(D) iterations Each iteration there suppose to be 1*P1+1*P2+ …+1*Pn =L points T otal Ln(D)*L points

16.Cluster the subset C Red points are in subset C

17.Cluster the sample C Calculate distances between point A to other points in C, and find the smallest distance In this case , d_c 1

18.Cluster the sample C Calculate distances between point A and all points in X, and get d_x i

19.Cluster the sample C Compare d_x i to d_c 1 , and let W A =number of d_x i < d_c 1 Then we get weight matrix, W Cluster W into k clusters, get k centers

20.Difference among three methods K means K means ++ K means || seeding choose k centers randomly Choose k centers proportionally Choose subset C and get k centers from C clustering

21.Hypothesis K means K means ++ K means || seeding choose k centers randomly fast Choose k centers proportionally slow Choose subset C and get k centers from C slower clustering

22.Hypothesis K means K means ++ K means || seeding choose k centers randomly fast Choose k centers proportionally slow Choose subset C and get k centers from C slower clustering slow fast faster

23.Test Hypothesis Toy data one – very small Cloud data – small Spam data – moderate Toy data two – very large

24.Simple data set N=100; d=2; k=2 ; Iteration=100

25.Executive time

26.Cloud data consists of 1024 points in 10 dimension k=6

27.

28.Executive time (in seconds)

29.Total scatter