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Committed GLL package v2.1
changes in v2.1 - demo_artificial_data_on_grid.m and additional functions to run this script are added
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‎README.txt‎

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Graph Laplacian Learning (GLL) Package v2.0.
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Graph Laplacian Learning (GLL) Package v2.1
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This MATLAB package includes implementations of graph learning algorithms presented in [1]-[2].
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To install the package:
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(1) Download the source files.
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(2) Run script 'start_graph_learning.m'
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(2) Run the script 'start_graph_learning.m'
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The demo script 'demo_animals.m' shows the usage of functions used to estimate three different graph Laplacian matrices discussed in [1].
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The demo script 'demo_us_temperature.m' shows the usage of functions used to estimate combinatorial Laplacian matrices from smooth signals discussed in [2]. The code regenerates Fig.7(e) in [2].
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The demo script 'demo_artificial_data_on_grid.m' implements the following steps:
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- generates a grid graph with random edges and a $\beta$-hop filter
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- generates an artificial dataset based on the graph system, which is specified by the generated graph and filter
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- runs the iterative algorithm described in [2].
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- shows ground truth and estimated graphs as well as returns the estimated $\beta$
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‎demo_artificial_data_on_grid.m‎

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clear
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clc
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close all
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%% Simulation parameters
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my_eps_outer = 10^(-4); my_eps_inner = 10^(-6);
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max_iterations = 20;
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%% Generate model
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% generate random 8x8 grid graph
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BS = 8;
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load(['Grid_' num2str(BS) 'x' num2str(BS) '.mat']);
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A_connectivity = grid_Con4; A_mask = A_connectivity;
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[Ltrue,~,~] = generateRandomGraphFromConnectivity(A_connectivity,3,0,'uniform');
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num_vertices = size(A_connectivity,2);
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% generate filter
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filter_type = 'hop'; beta = 3; % beta-hop filter with true beta parameter is set here
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graph_filter_ideal = @(x)(graph_filter_fwd(x,beta,filter_type) );
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% generate graph system
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[h_L,h_L_sqrMatrix] = generateFilteredModel(Ltrue,graph_filter_ideal);
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%% Generate data samples based on h_L
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S_data_cell = generateRandomDataFromSqrtM(h_L_sqrMatrix,30); % creates data with 30 samples per vertex
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S_data = S_data_cell{1}; S_data = 0.5*(S_data + S_data');
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%% Algorithm implementation
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beta_current = 1; % initialize beta
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% Eigendecompose data
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[U,sigma_sq_C] = createBasis(S_data,'descend');
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sigma_sq_C(sigma_sq_C <= 10^-10) = 0;
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for repeat=1:max_iterations
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disp(['-- Iteration ' num2str(repeat) '--']);
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% Step I: Prefiltering step
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lambdas_current = graph_filter_inv(sigma_sq_C,beta_current,filter_type);
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current_sigmas = 1./lambdas_current;
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current_sigmas(current_sigmas==Inf)=0;
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% construct unfiltered S
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S_prefiltered = U * diag(abs(current_sigmas)) * U';
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S_prefiltered = 0.5*(S_prefiltered + S_prefiltered'); % symmetrize (in case of numerical inaccuracies)
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max_eig_of_S_data = max(current_sigmas);
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S_prefiltered = S_prefiltered/max_eig_of_S_data; % normalize
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% Step II: Graph learning step
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Laplacian = estimate_cgl(S_prefiltered,ones(num_vertices),eps,my_eps_outer,my_eps_inner,max_iterations);
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Laplacian = Laplacian/max_eig_of_S_data; % normalize
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% Step III: Parameter estimation step
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estimated_lambdas = diag(U' * Laplacian * U); estimated_lambdas(estimated_lambdas <= 10^-10) = 0;
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estimated_sigmas = graph_filter_fwd(estimated_lambdas,repeat,filter_type);
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h_L_current = U * diag(estimated_sigmas) * U'; h_L_current = 0.5*(h_L_current + h_L_current');
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error_est(repeat) = norm(S_data-h_L_current,'fro')/norm(S_data,'fro');
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% convergence criterion
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if repeat > 1 && abs(error_est(repeat-1) - error_est(repeat)) > 0.2
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disp('*** Algorithm has converged ***');
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break;
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end
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beta_current = beta_current + 1;
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end
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disp(['Estimated beta = ' num2str(beta_current) ]);
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disp(' Figure 1 shows the ground truth graph');
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disp(' Figure 2 shows the estimated graph');
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f1=figure(1);
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draw_grid_graph(laplacianToAdjacency(Ltrue,eps),BS);
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axis square
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movegui(f1,'west');
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f2=figure(2);
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draw_grid_graph(laplacianToAdjacency(Laplacian,eps),BS);
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axis square
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movegui(f2,'east');
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‎functions/generateRandomDataFromSqrtM.m‎

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% Function for generating random data given square root matrix assoc. with graph
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%
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% (C) Hilmi Egilmez
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function [S_cell,D_cell] = generateRandomDataFromSqrtM(SquareRootMatrix,k_over_n_list)
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num_covars = length(k_over_n_list);
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S_cell = cell(1,num_covars); % cell of covariance matrices
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D_cell = cell(1,num_covars); % cell of sample distance matrices
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num_vertices = size(SquareRootMatrix,2);
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degree_of_freedom = num_vertices; %num_vertices*(num_vertices-1)/2;
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% degree_of_freedom = num_vertices*(num_vertices-1)/2;
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nSamplesList = int32(round(degree_of_freedom*k_over_n_list)); % number of data samples
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y_samples = SquareRootMatrix*randn(num_vertices,max(nSamplesList));
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for i=1:length(k_over_n_list)
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y_selected = y_samples(:,1:nSamplesList(i));
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S_temp = cov(y_selected',1);
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S_cell{i} = S_temp;
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end
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end
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‎functions/generateRandomGraphFromConnectivity.m‎

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% Function for generating random data given
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%
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% (C) Hilmi Egilmez May 2016
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function [Ltrue,S_true,S_sqrt_matrix] = generateRandomGraphFromConnectivity(A_connectivity,edge_weight_par,vertex_weight_par,type_weighting)
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if nargin < 4
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type_weighting = 'uniform';
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if nargin < 3
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vertex_weight_par = edge_weight_par;
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end
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end
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num_vertices = size(A_connectivity,2);
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random_Seed = 'shuffle';
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num_edges = sum(A_connectivity(:))/2;
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rng(random_Seed); % seed random number generator
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% apply random weights depending on weight_var
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if isequal(type_weighting,'gaussian')
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% Gaussian random
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weights = abs(randn(1,num_edges)*sqrt(edge_weight_par))+0.1; %0.01 + exprnd(weight_var,1,num_edges); % abs(randn(1,num_edges)*sqrt(weight_var));
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vertex_weights = abs(randn(1,num_vertices)*sqrt(vertex_weight_par))+0.1; % 0.01 + exprnd(weight_var,1,num_vertices); %abs(randn(1,num_vertices)*sqrt(weight_var));
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elseif isequal(type_weighting,'uniform')
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% Uniform
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weights = (rand(1,num_edges)*(edge_weight_par-0.1))+0.1; %0.01 + exprnd(weight_var,1,num_edges); % abs(randn(1,num_edges)*sqrt(weight_var));
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vertex_weights = (rand(1,num_vertices)*(vertex_weight_par-0.1))+0.1; % 0.01 + exprnd(weight_var,1,num_vertices); %abs(randn(1,num_vertices)*sqrt(weight_var));
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else
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error('Wrong input: type_weighting')
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end
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% create Laplacian
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B = convertAdjToIncidence(A_connectivity);
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if vertex_weight_par == 0
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Ltrue = B * diag(weights) * B';
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else
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Ltrue = B * diag(weights) * B' + diag(vertex_weights);
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end
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% Create DataSamples
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[U,E] = eig(Ltrue);
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% scale graph values when it has too large/small e.vals
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d_e = diag(E);
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%%% added to avoid determinant < 1 case for vassilis algorithm
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d_e(d_e < 10^-10)=1;
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det_L = prod(d_e);
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if det_L < 1
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E = E*(det_L^(-1/num_vertices)+eps);
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d_e = d_e*(det_L^(-1/num_vertices)+eps);
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end
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emin = min(d_e); emax = max(d_e);
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scaling = 1;
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if det_L == Inf || det_L==0
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scales = linspace(emin,emax,10);
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for i=1:length(scales)
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s = scales(i);
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det_L = prod(d_e./s);
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if det_L ~= Inf && det_L~=0
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scaling = s;
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break;
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end
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end
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end
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if det_L == Inf || det_L == 0
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error(['Function generateRandomGraphFromConnectivity : Determinant of Laplacian is equal to ' ...
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num2str(det_L) ' -- try different parameters to generate random graph weights' ]);
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end
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Ltrue = Ltrue/scaling;
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E = E/scaling;
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E(E<=10^(-10)) = 0;
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idx = E>10^(-10);
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E(idx) = abs(1./E(idx));
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S_true = U* E *U';
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%%% outputs
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S_sqrt_matrix = U*(E.^(0.5));
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S_true = 0.5*S_true + 0.5*S_true'; % symmetrize
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Ltrue = 0.5*Ltrue + 0.5*Ltrue';% symmetrize
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end

‎misc/Grid_8x8.mat‎

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‎misc/convertAdjToIncidence.m‎

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% Input:
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% Adj: 0/1 adjacency matrix (no self loops) given the *undirected* graph connection pattern
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% Output:
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% Incidence : Incidence matrix
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% fromNodes : list of *from* nodes
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% toNodes : list of *to* nodes
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function [Incidence,fromNodes,toNodes] = convertAdjToIncidence(Adj)
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% take lower triangular part
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upper = ~logical(tril(Adj));
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Adj(upper) = 0;
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% Find edges to design
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[fromNodes,toNodes] = find(Adj==1);
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nNodes = size(Adj,1);
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nLinks = length(fromNodes);
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% create Incidence Matrix
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Incidence = zeros(nNodes,nLinks);
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for l=1:nLinks
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i=fromNodes(l); j=toNodes(l);
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Incidence(i,l) = 1; Incidence(j,l) = -1;
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end
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end

‎misc/draw_grid_graph.m‎

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%% Draw a grid graph with link weights having different colors
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% Inputs: A: Adjacency matrix
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% BS: block size
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function draw_grid_graph(A,BS)
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load ColorMatrix;
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for i=1:BS
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for j=1:BS
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pixel = i+(j-1)*BS;
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graph_points(pixel,2)= -i; % height
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graph_points(pixel,1)= j; % width
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end
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end
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colors = colormap(Fire/255);
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wgPlot_grid(A,graph_points,'edgeColorMap',colors,'edgeWidth',2,'vertexColorMap',255*[1 1 1]);
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set(gcf,'color','w');
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% wgPlot(A,graph_points,'edgeColorMap',jet);
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% set(gcf,'color','w');

‎misc/laplacianToAdjacency.m‎

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function A = laplacianToAdjacency(L,epsilon)
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sizeMat = size(L);
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offDiag = ones(sizeMat) - eye(sizeMat); offDiag = offDiag == 1;
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A = zeros(sizeMat); A(offDiag) = -L(offDiag);
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A(A<epsilon) = 0;
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end

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