slpm_nga {SparseLPM}

slpm_nga

Description

Runs a Natural Gradient Ascent algorithm to maximise the variational objective for a Sparse LPM.

Usage

slpm_nga(X, K, var_pars_init, hyper_pars = NULL, tol = 0.01, n_iter_max = 100000,
        natural_gradient = T, learning_rate_factor_up = 2, learning_rate_factor_down = 2,
        verbose = F)

Arguments

X

Rectangular adjacency matrix with non-negative entries.

K

The number of latent dimension of the model.

var_pars_init

List of variational parameters to be used as starting point for the optimisation. See Details for more specific indications.

hyper_pars

List defining the hyperparameters of the model. The list should contain three vectors of K positive values denoted delta, a_gamma and b_gamma, respectively, where K is the number of latent dimensions. If left as null, all delta parameters are set to 0.001, whereas the rest is set to 1.

tol

Positive number setting the stop condition: the algorithm stops if one entire iteration yields an increase in the objective function smaller than this value.

n_iter_max

Maximum number of iterations the algorithm should be run for.

natural_gradient

TRUE or FALSE indicating whether the natural gradient instead of the standard gradient should be used.

learning_rate_factor_up

Before any natural gradient ascent update, the current step size is multiplied by this number to ensure that the algorithms tries new solutions which are relatively far from the current one.

learning_rate_factor_down

During any natural gradient ascent update, if a certain step size leads to a decrease in the objective function, then the step is divided by this number repeatedly until an increase is ensured.

verbose

TRUE or FALSE indicating whether a lengthy output should be printed out.

Details

var_pars and var_pars_init are lists with components:

alpha_u_tilde

M*K matrix representing the Gaussian means for the latent positions of senders.

alpha_v_tilde

N*K matrix representing the Gaussian means for the latent positions of receivers.

beta_u_tilde

M*K matrix representing the Gaussian variances for the latent positions of senders.

beta_v_tilde

N*K matrix representing the Gaussian variances for the latent positions of receivers.

lambda_tilde

M*N*K array with entries corresponding to the posterior probabilities of assigning each edge to each latent dimension.

delta_tilde

Vector of K positive values representing the Dirichlet parameters generating the mixing proportions.

a_tilde

Vector of K positive values corresponding to the shapes of the variational Gamma distribution on the precisions.

b_tilde

Vector of K positive values corresponding to the rates of the variational Gamma distribution on the precisions.

Value

A list with components:

computing_time

Number of seconds required for the optimisation process.

var_pars

List containing the optimal values for the variational parameters.

learning_rates_u

Current step-size for the update of the variational parameters of each Gaussian distribution on the latent positions of senders.

learning_rates_v

Current step-size for the update of the variational parameters of each Gaussian distribution on the latent positions of receivers.

elbo_values

Values of the variational objective at the end of each of the iterations.

elbo_init

Value of the variational objective for the initial configuration.

elbo_final

Value of the variational objective for the optimal solution found.

References

Rastelli, R. (2018) "The Sparse Latent Position Model for nonnegative weighted networks", https://arxiv.org/abs/1808.09262

Examples

set.seed(12345)
network <- slpm_gen(M = 15, N = 10, K = 2)
K <- 6
var_pars_init <- slpm_init(X = network$adj, K = K)
res <- slpm_nga(X = network$adj, K = K, var_pars_init = var_pars_init)