-
Notifications
You must be signed in to change notification settings - Fork 0
/
rate_distortion.m
268 lines (215 loc) · 9.51 KB
/
rate_distortion.m
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% Rate-distortion
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
setenv('LC_ALL','C');
warning off all;
close all
clear
clc
addpath(genpath('libs'));
% --- CONSTANTS -----------------------------------------------------------
n = 500; % number of measurement
M = 250; % number of samples
st = ceil(M/5); % sparsity changing step
S = [1:st:M]; % sparsity
t = length(S);
m = 10000;
% gradient descent params
T = 500; % number of iterations
is_all_iterations = 0; % 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 = 1000; % 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));
D = @(sx, R) sx*2^(-2*R); % rate-distortion
%% --- Prepare data -------------------------------------------------------
X_dbase = randn(m,n);
X_dbase = X_dbase - repmat(mean(X_dbase),m,1);
X = X_dbase;
%% --- Process ------------------------------------------------------------
D_upper_bound = zeros(Run, t);
D_map = zeros(Run, t);
D_pinv = zeros(Run, t);
D_gd_const_x_init_x0 = zeros(Run, t);
D_gd_const_x0_init_x0 = zeros(Run, t);
D_gd_const_x0_init_0 = zeros(Run, t);
D_gd_no_priors = zeros(Run, t);
D_amp_const_x_init_x0 = zeros(Run, t);
D_amp_const_x0_init_x0 = zeros(Run, t);
D_amp_const_x0_init_0 = zeros(Run, t);
D_amp_no_priors = zeros(Run, t);
R = zeros(1, t);
% -------------------------------------------------------------------------
for r=1:Run
i = 0;
A = randn(M, n); % transform matrix
if M < n
A = orth(A')';
else
A = orth(A);
end
x = randn(n, 1); % original signal
x = x - mean(x);
y = A*x;
x0 = x + randn(n, 1)*sigma; % side infromation
for s = S
i = i + 1;
R(i) = (log2(nchoosek(M,s)) + s)/n; % rate
DS(i) = var(x)*2^(-2*R(i)); % distortion
ys = sign(y).*kLagestMask(abs(y), s); % quantized-sparsified representation of x
% --- estimation of normalized coefficient for pseudo-inverse solution ----
[U, Sigm, V] = svd(A);
Ainv = V*pinv(Sigm)*U';
fX = X*A';
for ii=1:m
fX(ii, :) = sign(fX(ii, :)').*kLagestMask(abs(fX(ii, :)'), s);
end
X_hat = fX*(Ainv)';
b_ps = trace(X'*X_hat)/trace(X_hat'*X_hat); % normalized coefficient for pseudo-inverse solution
%--------------------------------------
% --- 1. Upper bound ----
lambda = var(y-ys)/var(x-x0); % Lagrangian multiplier
D_upper_bound(r, i) = rank(A)*norm( A'*(A*x-ys) + lambda*(x-x0) )^2/(n*(1+lambda));
% --- 2. MAP estimation ----
x_est = x0*var(x)/(var(x)+sigma^2);
D_map(r, i) = norm(x-x_est)^2/n;
% --- 3. pseudo-inverse ----
x_est = Ainv*ys;
D_pinv(r, i) = norm(x-x_est*b_ps)^2/n;
% --- 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 = 0; % 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, i) = norm(x-x_opt)^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 = 0; % 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, i) = norm(x-x_opt)^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 = 0; % 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, i) = norm(x-x_opt)^2/n;
% --- 5. single-layer gradient descent without side informaiton (lambda = 0)||Ax-y||^2
lambda = 0;
x_est = gradientDescentWithMomentum(zeros(size(x0)), x, A, ys, lambda, T, stepsize, gamma, ...
1, is_all_iterations);
D_gd_no_priors(r, i) = norm(x-x_est)^2/n;
% --- 6. AMP ----
% ----- 6.1 ||Ax-y||^2 + lambda ||x||^2 init via x0
lambda = var(y-ys)/var(x);
x_est = AMPConstrainOnX(x0, x, A, ys, lambda, T, stepsize, gamma, 0, is_all_iterations);
D_amp_const_x_init_x0(r, i) = norm(x-x_est)^2/n;
% --- 6.2 ||Ax-y||^2 + lambda ||x-x_0||^2 init via x0
lambda = var(y-ys)/var(x-x0);
x_est = AMP(x0, x, A, ys, lambda, T, stepsize, gamma, 0, is_all_iterations);
D_amp_const_x0_init_x0(r, i) = norm(x-x_est)^2/n;
% --- 6.3 AMP + lambda ||x-x_0||^2 init via 0
lambda = var(y-ys)/var(x-x0);
x_est = AMP(x0, x, A, ys, lambda, T, stepsize, gamma, 1, is_all_iterations);
D_amp_const_x0_init_0(r, i) = norm(x-x_est)^2/n;
% --- 7. AMP without side informaiton (lambda = 0)||Ax-y||^2
lambda = 0;
x_est = AMP(zeros(size(x0)), x, A, ys, lambda, T, stepsize, gamma, 1, is_all_iterations);
D_amp_no_priors(r, i) = norm(x-x_est)^2/n;
end
end
%% --- averaging ----------------------------------------------------------
D_upper_bound = mean(D_upper_bound, 1);
D_map = mean(D_map, 1);
D_pinv = mean(D_pinv, 1);
D_gd_const_x_init_x0 = mean(D_gd_const_x_init_x0, 1);
D_gd_const_x0_init_x0 = mean(D_gd_const_x0_init_x0, 1);
D_gd_const_x0_init_0 = mean(D_gd_const_x0_init_0, 1);
D_gd_no_priors = mean(D_gd_no_priors, 1);
D_amp_const_x_init_x0 = mean(D_amp_const_x_init_x0, 1);
D_amp_const_x0_init_x0 = mean(D_amp_const_x0_init_x0, 1);
D_amp_const_x0_init_0 = mean(D_amp_const_x0_init_0, 1);
D_amp_no_priors = mean(D_amp_no_priors, 1);
%% --- visualisation ------------------------------------------------------
lw = 2;
figure;
plot(linspace(R(1), max(4.5, max(R)),100), var(x)*2.^(-2*linspace(R(1),max(4.5, max(R)),100)), ...
'r', 'LineWidth', lw); hold on;
plot(R, D_map, '--k', 'LineWidth', lw); hold on;
plot(R, D_pinv, '--r', 'LineWidth', lw); hold on;
plot(R, D_gd_const_x_init_x0, 'b', 'LineWidth', lw); hold on;
plot(R, D_gd_const_x0_init_x0, 'g', 'LineWidth', lw); hold on;
plot(R, D_gd_const_x0_init_0, '--g', 'LineWidth', lw); hold on;
plot(R, D_gd_no_priors, '--b', 'LineWidth', lw); hold on;
plot(R, D_amp_const_x_init_x0, 'm', 'LineWidth', lw); hold on;
plot(R, D_amp_const_x0_init_x0, 'c', 'LineWidth', lw); hold on;
plot(R, D_amp_const_x0_init_0, '--c', 'LineWidth', lw); hold on;
plot(R, D_amp_no_priors, '--m', 'LineWidth', lw); hold on;
plot(R, D_upper_bound, 'k', 'LineWidth', lw); hold on;
grid on;
legend('shannon', 'MAP', 'pinv', ...
'GD ||Ax-y||^2 + lambda ||x||^2 init via x0', ...
'GD ||Ax-y||^2 + lambda ||x-xo||^2init via x0', ...
'GD ||Ax-y||^2 + lambda ||x-xo||^2init via 0', ...
'GD ||Ax-y||^2', ...
'AMP ||Ax-y||^2 + lambda ||x||^2 init via x0', ...
'AMP ||Ax-y||^2 + lambda ||x-xo||^2init via x0', ...
'AMP ||Ax-y||^2 + lambda ||x-xo||^2init via 0', ...
'AMP ||Ax-y||^2', ...
'upper bound' ...
);
xlabel('rate');
ylabel('distortion');
title(['n = ', num2str(n), ', ', ...
'M = ', num2str(M), ', ', ...
'sigma_z^2 = ', num2str(sigma^2)])
%%----------------------------------------------------------------------------