# Solve the internal minimization problem This internal function of the l1-spectral clustering algorithm solves the l1-minimization problem and recover the community indicators of the clusters. ```r PenOpt(U, n, elements, iteration, pen, k) ``` ## Arguments - `U`: The eigenvector matrix. - `n`: The number of nodes of the connected component to cluster. - `elements`: The representative elements of the connected component to cluster. - `iteration`: The cluster we aim at recovering. - `pen`: The penalty (to be chosen among "lasso" and "thresholdedLS"). - `k`: The number of clusters. ## Returns `v` The community indicator of cluster `iteration`. ## Examples ```r ################################### # Solving the minimization problem ################################### # 1st: create data Data <- CreateDataSet(k=3, n=20, p=list(p_inside=0.1,p_outside=0.1)) # 2nd: find the structure, the opt number of clusters and the representative elements Structure <- FindStructure(Data$A_hat) Clusters <- FindNbrClusters(A = Data$A_hat, structure = Structure) Elements <- FindElement(A = Data$A_hat, structure = Structure, clusters = Clusters) Structure_tmp <- Structure$groups[[1]] # the first component A_tmp <- Data$A_hat[Structure$groups[[1]],Structure$groups[[1]]] n <- ncol(A_tmp) k <- Clusters$nbr_clusters$Component1 # number of clusters to create Elements_tmp <- Elements$indices$Component1 # the elements of the first component # 3rd: perform svd svd <- eigen(A_tmp) eigenvalues <- sort(svd$values,index.return=TRUE) eigenvectors <- svd$vectors[,eigenvalues$ix] # 4th: solve the minimization problem i <- 1 # the cluster we aim at recovering U <- t(eigenvectors[,1:(n-k+i-1)]) v <- PenOpt(U, n, elements = Elements_tmp, iteration = i, pen = "lasso", k) # for lasso # the same with the least-squared threshold v <- PenOpt(U, n, elements = Elements_tmp, iteration = i, pen = "thresholdedLS", k) ``` ## See Also `l1_spectralclustering`, `l1_spectral`, `l1spectral`. ## Author(s) Camille Champion, Magali Champion

PenOpt function