# Nonparametric smooth approximation of the Laplacian graph spectra This function provides nonparametric smooth approximation of the Laplacian graph spectra given a weighted-adjacency matrix $W$. ```r LPSpectral(W, k, m=8,sparse=TRUE) ``` ## Arguments - `W`: A $n$-by-$n$ weighted-adjacency matrix. - `k`: Number of approximated singular vectors and singular values to return, where $k \leq m$. - `m`: Number of LP-nonparametric basis used for approximation, where $m \leq n$. By default, $m=8$. - `sparse`: Set to `TRUE` to make coefficients for LP basis sparse, thus allowing for further smoothing. ## Returns A list containing the following items: - **LP**: $m$-by-$m$ LP Spectral graph matrix. - **Phi**: A $n$-by-$k$ matrix of LP-approximated singular vectors. - **sval**: A vector of length $k$ containing top k approximated singular values. ## References Mukhopadhyay, S. and Wang, K. (2018), "Graph Spectral Compression via Smoothing". ## Author(s) Mukhopadhyay, S. and Wang, K. ## Examples ```r ##1.toy example: ##simulate a two sample locational difference normal data: X1<-matrix(rnorm(250,mean=0,sd=1),10,25) X2<-matrix(rnorm(250,mean=0.5,sd=1),10,25) X<-rbind(X1,X2) ## Adjacency matrix: dmat<-dist(X) W <-exp(-as.matrix(dmat)^2/(2*quantile(dmat,.5)^2)) ## Obtain top 10 approximated nontrivial singular values: data_sval<-LPSpectral(W, k=10)$sval ## Obtain approximated singular vector corresponding to the top nontrivial singular value: data_phi1<-LPSpectral(W, k=1)$Phi ## plot the results: par(mfrow=c(1,2)) plot(data_sval,type='b') plot(data_phi1) ##2.Senate Data ## Not run: data(senate) attach(senate) ##creating W (long computation) require(psych) W <- matrix(0,nrow(X),nrow(X)) for(i in 1:(nrow(X)-1)){ for(j in (i+1):nrow(X)) { W[i,j] <- psych::phi(table(X[i,],X[j,])) } } W = W + t(W) diag(W)<-0 ## Obtain top 10 approximated nontrivial singular values: senate_sval<-LPSpectral(W, k=10, m=15)$sval ## Obtain approximated singular vector corresponding to the top nontrivial singular value: senate_phi1<-LPSpectral(W, k=1, m=15)$Phi ## plot the results: par(mfrow=c(1,2)) plot(senate_sval,type='b') plot(senate_phi1) ## End(Not run) ```

LPSpectral function