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convergence_rate.m
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% Convergence rate of recovering the original signal x from a quantized
% sparse representation y=Q(phi(A)) with a given noisy observation
% x_0 = x + z
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
setenv('LC_ALL','C');
warning off all;
close all
clear
clc
addpath(genpath('libs'));
% --- CONSTANTS -----------------------------------------------------------
n = 500; % number of measurement
M = 500; % number of samples
s = ceil(0.09*M); % sparsity
m = 10000;
% gradient descent params
T = 500; % number of iterations
is_all_iterations = 1; % 1 - finish after T iteration; 0 - finish after convergence;
stepsize = 1; % step size
gamma = 0.005; % backtracking line search parameter
is_zero = 0; % start initialisation type: 0 - start from prior; 1 - strart from zero
Run = 3; % number of runs
sigma = 1; % noise standard deviation
% --- Functions -----------------------------------------------------------
thresholds = @(sortedData, K) sortedData(K, :);
kLagestMask = @(P, K) bsxfun(@ge,P, thresholds(sort(P, 'descend'), K));
%% --- Process ------------------------------------------------------------
X_dbase = randn(m,n); X_dbase = X_dbase - repmat(mean(X_dbase),m,1);
X = X_dbase;
%--------------------------------------------------------------------------
% --- estimation of normalized coefficient for pseudo-inverse solution ----
A = randn(M, n); % transform matrix
if M < n
A = orth(A')';
else
A = orth(A);
end
fX = X*A';
for ii=1:m
fX(ii, :) = sign(fX(ii, :)').*kLagestMask(abs(fX(ii, :)'), s);
end
[U, S, V] = svd(A);
X_hat = fX*(V*pinv(S)*U')';
b_ps = trace(X'*X_hat)/trace(X_hat'*X_hat);
%--------------------------------------------------------------------------
for r=1:Run
A = randn(M, n); % transform matrix
if M < n
A = orth(A')';
else
A = orth(A);
end
z = randn(n, 1)*sigma; % noise
x = randn(n, 1); % original signal
x0 = x + z; % side informaiton
y = A*x;
ys = sign(y).*kLagestMask(abs(y), s); % quantized-sparsified representation of x
sx = var(x); % variance of original signal
sz = sigma^2; % variance of noise
% --- 1. MAP estimation ----
x_est = x0*sx/(sx+sz);
D_map(r, :) = repmat(norm(x-x_est)^2/n, 1, T-1);
% --- 2. pseudo-inverse ----
x_est = b_ps*(pinv(A'*A)*A'*ys);
D_pinv(r, :) = repmat(norm(x-x_est)^2/n, 1, T-1);
% --- 4. multi-layer gradient descent ----
% ----- 4.1 ||Ax-y||^2 + lambda ||x||^2 init via x0
x_prior = x0; % side informaiton
j = 1; % number of layers
D_j = 0; % obtained distortion at eat layer
D_all = []; % obtained distortion at all layers
x_opt = zeros(size(x)); % optimal estimation
while j <= 2 || min(D_j) < min(D_all)
if j > 1
D_all = cat(2, D_all, D_j);
x_prior = x_est*var(x)/(var(x)+var(x-x_est));
end
lambda = var(y-ys)/var(x);
[x_est, D_j] = gradientDescentConstrainOnXWithMomentum(x_prior, x, A, ys, lambda, T, stepsize, ...
gamma, 0, is_all_iterations);
if norm(x-x_est) < norm(x-x_opt)
x_opt = x_est;
end
j = j + 1;
end
D_gd_const_x_init_x0(r, :) = (D_all.^2)./n;
% ----- 4.2. ||Ax-y||^2 + lambda ||x-x0||^2 init via x0
x_prior = x0; % side informaiton
j = 1; % number of layers
D_j = 0; % obtained distortion at eat layer
D_all = []; % obtained distortion at all layers
x_opt = zeros(size(x)); % optimal estimation
while j <= 2 || min(D_j) < min(D_all)
if j > 1
D_all = cat(2, D_all, D_j);
x_prior = x_est*var(x)/(var(x)+var(x-x_est));
end
lambda = var(y-ys)/var(x-x_prior);
[x_est, D_j] = gradientDescentWithMomentum(x_prior, x, A, ys, lambda, T, stepsize, ...
gamma, 0, is_all_iterations);
if norm(x-x_est) < norm(x-x_opt)
x_opt = x_est;
end
j = j + 1;
end
D_gd_const_x0_init_x0(r, :) = (D_all.^2)./n;
% ----- 4.3. ||Ax-y||^2 + lambda ||x-xo||^2 init via 0
x_prior = x0; % side informaiton
j = 1; % number of layers
D_j = 0; % obtained distortion at eat layer
D_all = []; % obtained distortion at all layers
x_opt = zeros(size(x)); % optimal estimation
while j <= 2 || min(D_j) < min(D_all)
if j > 1
D_all = cat(2, D_all, D_j);
x_prior = x_est*var(x)/(var(x)+var(x-x_est));
end
lambda = var(y-ys)/var(x-x_prior);
[x_est, D_j] = gradientDescentWithMomentum(x_prior, x, A, ys, lambda, T, stepsize, ...
gamma, 1, is_all_iterations);
if norm(x-x_est) < norm(x-x_opt)
x_opt = x_est;
end
j = j + 1;
end
D_gd_const_x0_init_0(r, :) = (D_all.^2)./n;
% --- 5. AMP ----
% ----- 5.1 ||Ax-y||^2 + lambda ||x||^2 init via x0
lambda = var(y-ys)/var(x);
[x_est, E] = AMPConstrainOnX(x0, x, A, ys, lambda, T, stepsize, gamma, 0, is_all_iterations);
D_amp_const_x_init_x0(r, :) = (E.^2)./n;
% --- 5.2 ||Ax-y||^2 + lambda ||x-x_0||^2 init via x0
lambda = var(y-ys)/var(x-x0);
[x_est, E] = AMP(x0, x, A, ys, lambda, T, stepsize, gamma, 0, is_all_iterations);
D_amp_const_x0_init_x0(r, :) = (E.^2)./n;
% --- 5.3 AMP + lambda ||x-x_0||^2 init via 0
lambda = var(y-ys)/var(x-x0);
[x_est, E] = AMP(x0, x, A, ys, lambda, T, stepsize, gamma, 1, is_all_iterations);
D_amp_const_x0_init_0(r, :) = (E.^2)./n;
end
%% --- averaging ----------------------------------------------------------
D_map = average(D_map);
D_pinv = average(D_pinv);
D_gd_const_x_init_x0 = average(D_gd_const_x_init_x0);
D_gd_const_x0_init_x0 = average(D_gd_const_x0_init_x0);
D_gd_const_x0_init_0 = average(D_gd_const_x0_init_0);
D_amp_const_x_init_x0 = average(D_amp_const_x_init_x0);
D_amp_const_x0_init_x0 = average(D_amp_const_x0_init_x0);
D_amp_const_x0_init_0 = average(D_amp_const_x0_init_0);
%% --- visualisation ------------------------------------------------------
lw = 2;
figure;
plot(D_map, 'k', 'LineWidth', lw); hold on;
plot(D_pinv, 'c', 'LineWidth', lw); hold on;
plot(D_gd_const_x_init_x0, 'b', 'LineWidth', lw); hold on;
plot(D_gd_const_x0_init_x0, 'r', 'LineWidth', lw); hold on;
plot(D_gd_const_x0_init_0, 'g', 'LineWidth', lw); hold on;
plot(D_amp_const_x_init_x0, '--b', 'LineWidth', lw); hold on;
plot(D_amp_const_x0_init_x0, '--r', 'LineWidth', lw); hold on;
plot(D_amp_const_x0_init_0, '--g', 'LineWidth', lw); hold on;
grid on;
legend('MAP(xo)', 'A^{+}y', ...
'GD: ||y-Ax||^2+lambda||x||^2 init via x0', ...
'GD: ||y-Ax||^2+lambda||x-xo||^2 init via x0', ...
'GD: ||y-Ax||^2+lambda||x-xo||^2 init via 0', ...
'AMP: ||y-Ax||^2+lambda||x||^2 init via x0', ...
'AMP: ||y-Ax||^2+lambda||x-xo||^2 init via x0', ...
'AMP: ||y-Ax||^2+lambda||x-xo||^2 init via 0' ...
);
xlabel('iteration');
ylabel('distortion');
title(['n = ', num2str(n), ', ', ...
'M = ', num2str(M), ', ', ...
'sigma_z^2 = ', num2str(sigma^2)]);