Two-sigma notation k-means clusteringI am looking at a k-means clustering algorithm, where the problem is the following: argmin C 1 , . . C k ; c 1 c 1 , . . c k c k ∑ i = 1 k ∑ x ∈ C i | | x x − c i c i | | 2 where x is a data vector, e.g. x 1 x 1 = ( 1 , 2 , 3 ) ′ , c i c i is the centroid of C i cluster. I conceptually understand what the above means, namely we find such centroids for each cluster that minimize the total squared distance between each centroid in a cluster and datapoints in that cluster. But I fail to understand how I would expand out the above expression.

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Two-sigma notation k-means clusteringI am looking at a k-means clustering algorithm, where the problem is the following:                    argmin                              C            1                    ,          .          .                      C            k                    ;                                                    c                1                                                                                  c              1                                ,          .          .                                                    c                k                                                                                  c              k                                                  ∑                  i          =          1                k                    ∑                  x          ∈                      C            i                                      |                    |                              x                                  x            −                                    c            i                                                      c          i                            |                              |                2            where x is a data vector, e.g.                     x        1                          x      1        =  (  1  ,  2  ,  3      )    ′  ,                     c        i                          c      i       is the centroid of       C    i   cluster. I conceptually understand what the above means, namely we find such centroids for each cluster that minimize the total squared distance between each centroid in a cluster and datapoints in that cluster. But I fail to understand how I would expand out the above expression.

parisjames70tiv · Accepted Answer

The idea of the double sigma notation is that its a sum of sums:      ∑          i      =      1        k        ∑          x      ∈              C        i              ‖  x  −      c    i        ‖    2    =      ∑          x      ∈              C        1              ‖  x  −      c    1        ‖    2    +  …  +      ∑          x      ∈              C        k              ‖  x  −      c    k        ‖    2  Each inner sum iterates over the data points of the according (current) Cluster       C    i  .

Two-sigma notation k-means clustering I am looking at a k-means clustering algorithm, where the pro

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