# Learn a bipartite graph Learns a bipartite graph on the basis of an observed data matrix ```r learn_bipartite_graph( S, is_data_matrix = FALSE, z = 0, nu = 10000, alpha = 0, w0 = "naive", m = 7, maxiter = 10000, abstol = 1e-06, reltol = 1e-04, record_weights = FALSE, verbose = TRUE ) ``` ## Arguments - `S`: either a pxp sample covariance/correlation matrix, or a pxn data matrix, where p is the number of nodes and n is the number of features (or data points per node) - `is_data_matrix`: whether the matrix S should be treated as data matrix or sample covariance matrix - `z`: the number of zero eigenvalues for the Adjancecy matrix - `nu`: regularization hyperparameter for the term ||A(w) - V Psi V'||^2_F - `alpha`: L1 regularization hyperparameter - `w0`: initial estimate for the weight vector the graph or a string selecting an appropriate method. Available methods are: "qp": finds w0 that minimizes ||ginv(S) - L(w0)||_F, w0 >= 0; "naive": takes w0 as the negative of the off-diagonal elements of the pseudo inverse, setting to 0 any elements s.t. w0 < 0 - `m`: in case is_data_matrix = TRUE, then we build an affinity matrix based on Nie et. al. 2017, where m is the maximum number of possible connections for a given node - `maxiter`: the maximum number of iterations - `abstol`: absolute tolerance on the weight vector w - `reltol`: relative tolerance on the weight vector w - `record_weights`: whether to record the edge values at each iteration - `verbose`: whether to output a progress bar showing the evolution of the iterations ## Returns A list containing possibly the following elements: - **`laplacian`**: the estimated Laplacian Matrix - **`adjacency`**: the estimated Adjacency Matrix - **`w`**: the estimated weight vector - **`psi`**: optimization variable accounting for the eigenvalues of the Adjacency matrix - **`V`**: eigenvectors of the estimated Adjacency matrix - **`elapsed_time`**: elapsed time recorded at every iteration - **`convergence`**: boolean flag to indicate whether or not the optimization converged - **`obj_fun`**: values of the objective function at every iteration in case record_objective = TRUE - **`negloglike`**: values of the negative loglikelihood at every iteration in case record_objective = TRUE - **`w_seq`**: sequence of weight vectors at every iteration in case record_weights = TRUE ## Examples ```r library(spectralGraphTopology) library(igraph) library(viridis) library(corrplot) set.seed(42) n1 <- 10 n2 <- 6 n <- n1 + n2 pc <- .9 bipartite <- sample_bipartite(n1, n2, type="Gnp", p = pc, directed=FALSE) # randomly assign edge weights to connected nodes E(bipartite)$weight <- runif(gsize(bipartite), min = 0, max = 1) # get true Laplacian and Adjacency Ltrue <- as.matrix(laplacian_matrix(bipartite)) Atrue <- diag(diag(Ltrue)) - Ltrue # get samples Y <- MASS::mvrnorm(100 * n, rep(0, n), Sigma = MASS::ginv(Ltrue)) # compute sample covariance matrix S <- cov(Y) # estimate Adjacency matrix graph <- learn_bipartite_graph(S, z = 4, verbose = FALSE) graph$adjacency[graph$adjacency < 1e-3] <- 0 # Plot Adjacency matrices: true, noisy, and estimated corrplot(Atrue / max(Atrue), is.corr = FALSE, method = "square", addgrid.col = NA, tl.pos = "n", cl.cex = 1.25) corrplot(graph$adjacency / max(graph$adjacency), is.corr = FALSE, method = "square", addgrid.col = NA, tl.pos = "n", cl.cex = 1.25) # build networks estimated_bipartite <- graph_from_adjacency_matrix(graph$adjacency, mode = "undirected", weighted = TRUE) V(estimated_bipartite)$type <- c(rep(0, 10), rep(1, 6)) la = layout_as_bipartite(estimated_bipartite) colors <- viridis(20, begin = 0, end = 1, direction = -1) c_scale <- colorRamp(colors) E(estimated_bipartite)$color = apply( c_scale(E(estimated_bipartite)$weight / max(E(estimated_bipartite)$weight)), 1, function(x) rgb(x[1]/255, x[2]/255, x[3]/255)) E(bipartite)$color = apply(c_scale(E(bipartite)$weight / max(E(bipartite)$weight)), 1, function(x) rgb(x[1]/255, x[2]/255, x[3]/255)) la = la[, c(2, 1)] # Plot networks: true and estimated plot(bipartite, layout = la, vertex.color=c("red","black")[V(bipartite)$type + 1], vertex.shape = c("square", "circle")[V(bipartite)$type + 1], vertex.label = NA, vertex.size = 5) plot(estimated_bipartite, layout = la, vertex.color=c("red","black")[V(estimated_bipartite)$type + 1], vertex.shape = c("square", "circle")[V(estimated_bipartite)$type + 1], vertex.label = NA, vertex.size = 5) ``` ## References S. Kumar, J. Ying, J. V. M. Cardoso, D. P. Palomar. A unified framework for structured graph learning via spectral constraints. Journal of Machine Learning Research, 2020. http://jmlr.org/papers/v21/19-276.html ## Author(s) Ze Vinicius and Daniel Palomar

learn_bipartite_graph function