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iCAPs/packages/gspbox/learn_graph/gsp_learn_graph_l2_degrees.m
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gsp_learn_graph_l2_degrees.m
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function [W, stat] = gsp_learn_graph_l2_degrees(Z, a, params)
%GSP_LEARN_GRAPH_L2_DEGREES Learn graph from pairwise distances using l2 prior on nodes degrees
% Usage: [W, stat] = gsp_learn_graph_l2_degrees(Z, a)
% [W, stat] = gsp_learn_graph_l2_degrees(Z, a, params)
%
% Inputs:
% Z : Matrix with (squared) pairwise distances of nodes
% a : L||_F^2 prior constant (bigger a -> more dense W)
% params : Optional parameters
%
% Outputs:
% W : Weighted adjacency matrix
% stat : Optional output statistics (adds small overhead)
%
% 'W = gsp_learn_graph_l2_degrees(Z, a, params)' computes a weighted
% adjacency matrix W from squared pairwise distances in Z, using the
% smoothness assumption that text{trace}(X^TLX) is small, where X is
% the data (columns) changing smoothly from node to node on the graph and
% L = D-W is the combinatorial graph Laplacian.
%
% Alternatively, Z can contain other types of distances and use the
% smoothness assumption that
%
% sum(sum(W .* Z))
%
% is small.
%
% The minimization problem solved is
%
% minimize_W sum(sum(W .* Z)) + a/2 * ||W||_F^2/2
% + a/2 * ||sum(W)||_2^2
% s.t. sum(sum(W)) == n
%
% subject to W being a valid weighted adjacency matrix (non-negative,
% symmetric, with zero diagonal). Note that
%
% .||W||_F^2/2 + ||sum(W)||_2^2 = ||L||^2
%
% The algorithm used is forward-backward-forward (FBF) based primal dual
% optimization (see references).
%
% Example:
%
% G = gsp_sensor(256);
% f1 = @(x,y) sin((2-x-y).^2);
% f2 = @(x,y) cos((x+y).^2);
% f3 = @(x,y) (x-.5).^2 + (y-.5).^3 + x - y;
% f4 = @(x,y) sin(3*((x-.5).^2+(y-.5).^2));
% X = [f1(G.coords(:,1), G.coords(:,2)), f2(G.coords(:,1), G.coords(:,2)), f3(G.coords(:,1), G.coords(:,2)), f4(G.coords(:,1), G.coords(:,2))];
% figure; subplot(2,2,1); gsp_plot_signal(G, X(:,1)); title('1st smooth signal');
% subplot(2,2,2); gsp_plot_signal(G, X(:,2)); title('2nd smooth signal');
% subplot(2,2,3); gsp_plot_signal(G, X(:,3)); title('3rd smooth signal');
% subplot(2,2,4); gsp_plot_signal(G, X(:,4)); title('4th smooth signal');
% Z = gsp_distanz(X').^2;
% [W] = gsp_learn_graph_l2_degrees(Z*25, 1);
% W(W<1e-5) = 0;
% G2 = gsp_update_weights(G, W);
% figure; gsp_plot_graph(G2); title('Graph with edges learned from above 4 signals');
%
%
% Additional parameters
% ---------------------
%
% params.W_init : Initialization point. default: zeros(size(Z))
% verbosity : Above 1 adds a small overhead. Default: 1
% maxit : Maximum number of iterations. Default: 1000
% tol : Tolerance for stopping criterion. Default: 1e-5
% step_size : Step size from the interval (0,1). Default: 0.5
%
% The stopping criterion is whether both relative primal and dual
% distance between two iterations are below a given tolerance.
%
% To set the step size use the following rule of thumb: Set it so that
% relative change of primal and dual converge with similar rates (use
% verbosity > 1).
%
%
% See also: gsp_learn_graph_log_degrees gsp_distanz gsp_update_weights
%
% References:
% V. Kalofolias. How to learn a graph from smooth signals. Technical
% report, AISTATS 2016: proceedings at Journal of Machine Learning
% Research (JMLR)., 2016.
%
% N. Komodakis and J.-C. Pesquet. Playing with duality: An overview of
% recent primal? dual approaches for solving large-scale optimization
% problems. Signal Processing Magazine, IEEE, 32(6):31--54, 2015.
%
%
%
% Url: http://lts2research.epfl.ch/gsp/doc/learn_graph/gsp_learn_graph_l2_degrees.php
% Copyright (C) 2013-2016 Nathanael Perraudin, Johan Paratte, David I Shuman.
% This file is part of GSPbox version 0.7.0
%
% This program is free software: you can redistribute it and/or modify
% it under the terms of the GNU General Public License as published by
% the Free Software Foundation, either version 3 of the License, or
% (at your option) any later version.
%
% This program is distributed in the hope that it will be useful,
% but WITHOUT ANY WARRANTY; without even the implied warranty of
% MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
% GNU General Public License for more details.
%
% You should have received a copy of the GNU General Public License
% along with this program. If not, see <http://www.gnu.org/licenses/>.
% If you use this toolbox please kindly cite
% N. Perraudin, J. Paratte, D. Shuman, V. Kalofolias, P. Vandergheynst,
% and D. K. Hammond. GSPBOX: A toolbox for signal processing on graphs.
% ArXiv e-prints, Aug. 2014.
% http://arxiv.org/abs/1408.5781
% Author: Vassilis Kalofolias
% Testing: gsp_test_learn_graph
% Date: June 2015
%% Default parameters
if nargin < 3
params = struct;
end
if not(isfield(params, 'verbosity')), params.verbosity = 1; end
if not(isfield(params, 'maxit')), params.maxit = 1000; end
if not(isfield(params, 'tol')), params.tol = 1e-5; end
if not(isfield(params, 'step_size')), params.step_size = .5; end % from (0, 1)
%% Fix parameter size and initialize
z = squareform(Z);
z = full(z(:));
l = length(z); % number of edges
% n(n-1)/2 = l => n = (1 + sqrt(1+8*l))/ 2
n = round((1 + sqrt(1+8*l))/ 2); % number of nodes
if isfield(params, 'W_init')
w = squareform(params.W_init)';
else
% w_0 = exp(-6*dx2/mean(dx2));
w = zeros(size(z));
end
%% Needed operators
% needed for gradient of h(w)
% L*w = sum(W)
[S, St] = sum_squareform(n);
% ones_l = ones(l,1);
sum_op = @(w) S*w;
sum_t_op = @(z) St*z;
% edges -> scalar
K_op = @(w) 2*sum(w); % sum(sum(W)) = 2 * sum(w)
% scalar -> edges
% Kt_op = @(z) 2 * z * ones_l;
norm_K = 2 * sqrt(l);
%norm_L = l; % do i need the square norm or the square root?
%% TODO: Rescaling??
%% Learn the graph
% min_{W>=0} tr(X'*L*X) - gc * sum(log(sum(W))) + gp * norm(W-W0,'fro')^2, where L = diag(sum(W))-W
% min_W I{W>=0} + W(:)'*Dx(:) - gc * sum(log(sum(W))) + gp * norm(W-W0,'fro')^2
% min_W f(W) + g(L_op(W)) + h(W)
% put proximal of trace plus positivity together
f.eval = @(w) 2*w'*z; % half should be counted
%f.eval = @(W) 0;
f.prox = @(w, c) max(0, w - 2*c*z); % all change the same
% projection of sum of W on n
% g(w) = IND(sum(w) == n)
g.eval = @(z) 10 * abs(z-n); % this could be 0. Measures violation of the condition sum(w) = n!!
g.prox = @(z, c) n;
% proximal of conjugate of g: z - c*g.prox(z/c, 1/c)
g_star_prox = @(z, c) z - c*n;
% the following is the squared frobenius norm of the graph Laplacian
h.eval = @(w) a * (2*norm(w)^2 + norm(sum_op(w))^2); % CAREFUL: w two times!
h.grad = @(w) 2 * a * (2*w + sum_t_op(sum_op(w)));
% if the next is wrong we don't converge for time_step != 0.5
h.beta = 2*(2*a*(n+1)); % = norm(ones(n)+eye(n)) = n+1
% TODO: best convergence when h.beta = 1 ?? then use rescaling!!
% if there is no quadratic term, it reduces to a linear program:
if a == 0
% X = linprog(f, A, b, Aeq, beq,LB,UB,X0)
[w, stat.fval, stat.exitflag, stat.output, stat.lambda] = linprog(z, [], [], ones(size(z))', 100*n, zeros(size(z)));
else
%% My custom FBF based primal dual (see [Komodakis, Pesquet])
% parameters mu, epsilon for convergence (see [1])
mu = h.beta + norm_K; %TODO: is it squared or not??
epsilon = lin_map(0.0, [0, 1/(1+mu)], [0,1]); % in (0, 1/(1+mu) )
% INITIALIZATION
% primal variable ALREADY INITIALIZED
%w = params.w_init;
% dual variable
v_n = K_op(w);
if nargout > 1 || params.verbosity > 1
stat.f_eval = nan(params.maxit, 1);
stat.g_eval = nan(params.maxit, 1);
stat.h_eval = nan(params.maxit, 1);
stat.fgh_eval = nan(params.maxit, 1);
stat.pos_violation = nan(params.maxit, 1);
end
if params.verbosity > 1
fprintf('Relative change of primal, dual variables, and objective fun\n');
end
tic
gn = lin_map(params.step_size, [epsilon, (1-epsilon)/mu], [0,1]); % in [epsilon, (1-epsilon)/mu]
for i = 1:params.maxit
%Y_n = w - gn * (h.grad(w) + Kt_op(v_n));
Y_n = w - gn * (h.grad(w) + 2*v_n);
y_n = v_n + gn * (K_op(w));
P_n = f.prox(Y_n, gn);
p_n = g_star_prox(y_n, gn); % = y_n - gn*g_prox(y_n/gn, 1/gn)
%Q_n = P_n - gn * (h.grad(P_n) + Kt_op(p_n));
Q_n = P_n - gn * (h.grad(P_n) + 2*p_n);
q_n = p_n + gn * (K_op(P_n));
if nargout > 1 || params.verbosity > 2
stat.f_eval(i) = f.eval(w);
stat.g_eval(i) = g.eval(K_op(w));
stat.h_eval(i) = h.eval(w);
stat.fgh_eval(i) = stat.f_eval(i) + stat.g_eval(i) + stat.h_eval(i);
stat.pos_violation(i) = -sum(min(0,w));
end
rel_norm_primal = norm(- Y_n + Q_n, 'fro')/norm(w, 'fro');
rel_norm_dual = norm(- y_n + q_n)/norm(v_n);
if params.verbosity > 2
fprintf('iter %4d: %6.4e %6.4e %6.4e %6.4e %6.3e \n', i, rel_norm_primal, rel_norm_dual, stat.fgh_eval(i), full(sum(w)*2), stat.h_eval(i));
elseif params.verbosity > 1
fprintf('iter %4d: %6.4e %6.4e\n', i, rel_norm_primal, rel_norm_dual);
end
if i <= 1 && isfield(params, 'W_init')
w = squareform(params.W_init)';
else
w = w - Y_n + Q_n;
end
v_n = v_n - y_n + q_n;
if rel_norm_primal < params.tol && rel_norm_dual < params.tol
break
end
end
stat.time = toc;
if params.verbosity > 0
fprintf('# iters: %4d. Rel primal: %6.4e Rel dual: %6.4e %6.3e\n', i, rel_norm_primal, rel_norm_dual, f.eval(w) + g.eval(K_op(w)) + h.eval(w));
fprintf('Time needed is %f seconds\n', stat.time);
end
% Use the following for testing:
% g.L = K_op;
% g.Lt = Kt_op;
% g.norm_L = norm_K;
% [w, info] = fbf_primal_dual(w, f, g, h, params);
%[w, info] = fb_based_primal_dual(w, f, g, h, params);
%%
if params.verbosity > 3
figure; plot(real([stat.f_eval, stat.g_eval, stat.h_eval])); hold all; plot(real(stat.fgh_eval), '.'); legend('f = w''*dx2', 'g = 1000 * abs(2sum(w)-n)', 'h = a*norm(L)^2', 'f+g+h');
figure; plot(stat.pos_violation); title('sum of negative (invalid) values per iteration')
figure; semilogy(max(0,-diff(real(stat.fgh_eval'))),'b.-'); hold on; semilogy(max(0,diff(real(stat.fgh_eval'))),'ro-'); title('|f(i)-f(i-1)|'); legend('going down','going up');
end
end
W = squareform(w);
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