SCG Problem Solver
This function solves the Spectral Coarse Graining (SCG) problem; either exactly, or approximately but faster.
scgGrouping(V, nt, mtype = c("symmetric", "laplacian", "stochastic"), algo = c("optimum", "interv_km", "interv","exact_scg"), p = NULL, maxiter = 100)
- A numeric matrix of (eigen)vectors to be preserved by the
coarse graining (the vectors are to be stored column-wise in
- A vector of positive integers of length one or equal to
ntcontains the number of groups used to partition each eigenvector separately. When
- The type of semi-projectors used in the SCG. For now
symmetric, laplacianand stochasticare available.
- The algorithm used to solve the SCG problem. Possible
optimum, interv_km, intervand exact_scg.
- A probability vector of length equal to
pis the stationary probability distribution of a Markov chain when
stochastic. This parameter is ignored in all other cases.
- A positive
integer giving the maximum number of iterations of the k-means
interv_km. This parameter is ignored in all other cases.
V. The running time of this algorithm is
$O(\max nt \cdot m^2)$ for the symmetric and
laplacian matrix problems (i.e. when
In all three cases, the memory usage is $O(m^2)$.
constant-size bins are used to partition
Once a minimizing partition (either exact or approximate) has been
found for each eigenvector, the final grouping is worked out as
follows: two vertices are grouped together in the final partition if
they are grouped together in each minimizing partition. In general the
size of the final partition is not known in advance when
Finally, the algorithm
- A vector of
nrow(V)integers giving the group label of each object (vertex) in the partition.
D. Morton de Lachapelle, D. Gfeller, and P. De Los Rios, Shrinking
Matrices while Preserving their Eigenpairs with Application to the
Spectral Coarse Graining of Graphs. Submitted to SIAM Journal on
Matrix Analysis and Applications, 2008.
# eigenvectors of a random symmetric matrix M <- matrix(rexp(10^6), 10^3, 10^3) M <- (M + t(M))/2 V <- eigen(M, symmetric=TRUE)$vectors[,c(1,2)] # displays size of the groups in the final partition gr <- scgGrouping(V, nt=c(2,3)) col <- rainbow(max(gr)) plot(table(gr), col=col, main="Group size", xlab="group", ylab="size") ## comparison with the grouping obtained by kmeans ## for a partition of same size gr.km <- kmeans(V,centers=max(gr), iter.max=100, nstart=100)$cluster op <- par(mfrow=c(1,2)) plot(V[,1], V[,2], col=col[gr], main = "SCG grouping", xlab = "1st eigenvector", ylab = "2nd eigenvector") plot(V[,1], V[,2], col=col[gr.km], main = "K-means grouping", xlab = "1st eigenvector", ylab = "2nd eigenvector") par(op) ## kmeans disregards the first eigenvector as it ## spreads a much smaller range of values than the second one ### comparing optimal and k-means solutions ### in the one-dimensional case. x <- rexp(2000, 2) gr.true <- scgGrouping(cbind(x), 100) gr.km <- kmeans(x, 100, 100, 300)$cluster scgNormEps(cbind(x), gr.true) scgNormEps(cbind(x), gr.km)