learn_bipartite_k_component_graph {spectralGraphTopology}

Learns a bipartite k-component graph Jointly learns the Laplacian and Adjacency matrices of a graph on the basis of an observed data matrix

Description

Learns a bipartite k-component graph

Jointly learns the Laplacian and Adjacency matrices of a graph on the basis of an observed data matrix

Usage

learn_bipartite_k_component_graph(
  S,
  is_data_matrix = FALSE,
  z = 0,
  k = 1,
  w0 = "naive",
  m = 7,
  alpha = 0,
  beta = 10000,
  rho = 0.01,
  fix_beta = TRUE,
  beta_max = 1e+06,
  nu = 10000,
  lb = 0,
  ub = 10000,
  maxiter = 10000,
  abstol = 1e-06,
  reltol = 1e-04,
  eigtol = 1e-09,
  record_weights = FALSE,
  record_objective = 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

k

the number of components of the graph

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

alpha

L1 regularization hyperparameter

beta

regularization hyperparameter for the term ||L(w) - U Lambda U'||^2_F

rho

how much to increase (decrease) beta in case fix_beta = FALSE

fix_beta

whether or not to fix the value of beta. In case this parameter is set to false, then beta will increase (decrease) depending whether the number of zero eigenvalues is lesser (greater) than k

beta_max

maximum allowed value for beta

nu

regularization hyperparameter for the term ||A(w) - V Psi V'||^2_F

lb

lower bound for the eigenvalues of the Laplacian matrix

ub

upper bound for the eigenvalues of the Laplacian matrix

maxiter

the maximum number of iterations

abstol

absolute tolerance on the weight vector w

reltol

relative tolerance on the weight vector w

eigtol

value below which eigenvalues are considered to be zero

record_weights

whether to record the edge values at each iteration

record_objective

whether to record the objective function 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

lambda

optimization variable accounting for the eigenvalues of the Laplacian matrix

V

eigenvectors of the estimated Adjacency matrix

U

eigenvectors of the estimated Laplacian matrix

elapsed_time

elapsed time recorded at every iteration

beta_seq

sequence of values taken by beta in case fix_beta = FALSE

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)
w <- c(1, 0, 0, 1, 0, 1) * runif(6)
Laplacian <- block_diag(L(w), L(w))
Atrue <- diag(diag(Laplacian)) - Laplacian
bipartite <- graph_from_adjacency_matrix(Atrue, mode = "undirected", weighted = TRUE)
n <- ncol(Laplacian)
Y <- MASS::mvrnorm(40 * n, rep(0, n), MASS::ginv(Laplacian))
graph <- learn_bipartite_k_component_graph(cov(Y), k = 2, beta = 1e2, nu = 1e2, verbose = FALSE)
graph$adjacency[graph$adjacency < 1e-2] <- 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)
# Plot networks
estimated_bipartite <- graph_from_adjacency_matrix(graph$adjacency, mode = "undirected",
                                                   weighted = TRUE)
V(bipartite)$type <- rep(c(TRUE, FALSE), 4)
V(estimated_bipartite)$type <- rep(c(TRUE, FALSE), 4)
la = layout_as_bipartite(estimated_bipartite)
colors <- viridis(20, begin = 0, end = 1, direction = -1)
c_scale <- colorRamp(colors)
E(estimated_bipartite)$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)