learn_bipartite_graph {spectralGraphTopology}

Learn a bipartite graph Learns a bipartite graph on the basis of an observed data matrix

Description

Learn a bipartite graph

Learns a bipartite graph on the basis of an observed data matrix

Usage

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

Value

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

Author(s)

Ze Vinicius and Daniel Palomar

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

Examples

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)