learn_bipartite_k_component_graph: Learns a bipartite k-component graph Jointly learns the...

In spectralGraphTopology: Learning Graphs from Data via Spectral Constraints

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)

spectralGraphTopology documentation built on March 18, 2022, 7:35 p.m.