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An alternative to normalized cut, called ratio cut, and shows that the methods of previous chapters can be adapted to solve the clustering problem using ratio cuts. The key idea is to replace the normalized laplacian with the unnormalized laplacian and omit the step of considering the problem (⇤⇤2) and multiplying the continuous solution y by d−1/2. The ratio cut is defined as the sum of the cut between each cluster and its complement, divided by the size of the cluster. Two formulations of the k-way clustering problem using ratio cuts, and discusses the natural relaxation of the second formulation. The algorithm is then extended to the case of signed graphs, where the signed ratio cut is defined. The document concludes by noting that normalized cut seems to yield 'better clusters', but that this statement is not very satisfactory since the precise definition of a 'better' clustering is left as further research.
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In this short chapter, we consider the alternative to nor- malized cut, called ratio cut, and show that the methods of Chapters 3 and 4 can be trivially adapted to solve the clustering problem using ratio cuts.
All that needs to be done is to replace the normalized Laplacian L (^) sym by the unormalized Laplacian L, and omit the step of considering Problem (⇤⇤ 2 ).
In particular, there is no need to multiply the continuous solution Y by D ^1 /^2.
201
202 CHAPTER 5. GRAPH CLUSTERING USING RATIO CUTS
The idea of ratio cut is to replace the volume vol(A (^) j ) of each block A (^) j of the partition by its size |A (^) j | (the number of nodes in Aj ).
First, we deal with unsigned graphs, the case where the entries in the symmetric weight matrix W are nonnega- tive.
Definition 5.1. The ratio cut Rcut(A 1 ,... , A (^) K ) of the partition (A 1 ,... , A (^) K ) is defined as
Rcut(A 1 ,... , A (^) K ) =
i=
cut(A (^) j , A (^) j ) |A (^) j |
204 CHAPTER 5. GRAPH CLUSTERING USING RATIO CUTS
On the other hand, the normalized cut is given by
Ncut(A 1 ,... , A (^) K ) =
i=
cut(A (^) j , A (^) j ) vol(A (^) j )
i=
(X j^ ) >^ LX j (X j^ ) >^ DX j^
Therefore, ratio cut is the special case of normalized cut where D = I.
If we let
n [X 1... X K^ ] | X j^ = a (^) j (x j 1 ,... , x jN ), x ji 2 { 1 , 0 },
a (^) j 2 R, X j^6 = 0
o
(note that the condition X j^6 = 0 implies that aj 6 = 0), then the set of matrices representing partitions of V into K blocks is
205
K =
n X = [X 1 · · · X K^ ] | X 2 X , (X i^ ) >^ X j^ = 0, 1 i, j K, i 6 = j
o .
Here is our first formulation of K-way clustering of a graph using ratio cuts, called problem PRC1 :
K-way Clustering of a graph using Ratio Cut, Version 1: Problem PRC
minimize
j=
(X j^ ) >^ LX j (X j^ ) >^ X j
subject to (X i^ ) >^ X j^ = 0, 1 i, j K, i 6 = j, X 2 X.
207
The natural relaxation of problem PRC2 is to drop the condition that X 2 X , and we obtain the
Problem (R⇤ 2 )
minimize tr(X >^ LX) subject to X >^ X = I.
This time, since the normalization condition is X >^ X = I, we can use the eigenvalues and the eigenvectors of L, and by Proposition A.2, the minimum is achieved by any K unit eigenvectors (u 1 ,... , u (^) K ) associated with the small- est K eigenvalues
0 = 1 2 ... (^) K
of L.
208 CHAPTER 5. GRAPH CLUSTERING USING RATIO CUTS
The matrix Z = Y = [u 1 ,... , u (^) K ] yields a minimum of our relaxed problem (R⇤ 2 ).
The rest of the algorithm is as before; we try to find Q = R⇤ with R 2 O(K), ⇤ diagonal invertible, and X 2 X such that kX ZQk is minimum.
In the case of signed graphs, we define the signed ratio cut sRcut(A 1 ,... , A (^) K ) of the partition (A 1 ,... , A (^) K ) as
sRcut(A 1 ,... , A (^) K ) =
j=
cut(Aj , A (^) j ) |A (^) j |
j=
links ^ (A (^) j , Aj ) |A (^) j |
210 CHAPTER 5. GRAPH CLUSTERING USING RATIO CUTS
Modifying the computer program implementing normal- ized cuts to deal with ratio cuts is trivial (use L instead
of L (^) sym and don’t multiply Y by D ^1 /^2 ).
Generally, normalized cut seems to yield “better clus- ters,” but this is not a very satisfactory statement since we haven’t defined precisely in which sense a clustering is better than another.
We leave this point as further research.
