Homework 5, Name: Yu Zheng

1. Spectrum analysis with quantized measurements

Code

qmax = []
for i = 1:n
    cvx_begin()
        variable q(n)
        maximize q(i)
        q >= 0;
        S*q >= l;
        S*q <= u;
    cvx_end()
    qmax = [qmax q(i)];
end
qmin = []
for i = 1:n
    cvx_begin()
        variable q(n)
        minimize q(i)
        q >= 0;
        S*q <= u;
        S*q >= l;
    cvx_end()
    qmin = [qmin q(i)];
end
figure; hold on;
for i = 1:n
    plot([i,i],[qmax(i),qmin(i)],'o-');
end
axis([0,11,0,1]);

Figure

2. Estimating a vector with unknown measurement nonlinearity.

Code

cvx_begin
    variables x(n) z(m)
    minimize sum(square(A * x - z))
    z(2:end)-z(1:end-1) >= (y(2:end) - y(1:end-1)) / beta
    z(2:end)-z(1:end-1) <= (y(2:end) - y(1:end-1)) / alpha
cvx_end
disp('x_ml = ')
disp(x)
figure()
plot(z,y)
title('\phi _{ml}')
xlabel('z')
ylabel('y')
grid on

Result

x_ml = 
    0.4819
   -0.4657
    0.9364
    0.9297

Figure

3.Three-way linear classification.

clear all; close all;
% three way separation problem 
% ee364a
% convex optimization

% data for problem instance
M = 20;
N = 20;
P = 20;

X = [

    3.5674    4.1253    2.8535    5.1892    4.3273    3.8133    3.4117 ...
    3.8636    5.0668    3.9044    4.2944    4.7143    3.3082    5.2540 ...
    2.5590    3.6001    4.8156    5.2902    5.1908    3.9802 ;...
   -2.9981    0.5178    2.1436   -0.0677    0.3144    1.3064    3.9297 ...
    0.2051    0.1067   -1.4982   -2.4051    2.9224    1.5444   -2.8687 ...
    1.0281    1.2420    1.2814    1.2035   -2.1644   -0.2821];

Y = [
   -4.5665   -3.6904   -3.2881   -1.6491   -5.4731   -3.6170   -1.1876 ...
   -1.0539   -1.3915   -2.0312   -1.9999   -0.2480   -1.3149   -0.8305 ...
   -1.9355   -1.0898   -2.6040   -4.3602   -1.8105    0.3096; ...
    2.4117    4.2642    2.8460    0.5250    1.9053    2.9831    4.7079 ...
    0.9702    0.3854    1.9228    1.4914   -0.9984    3.4330    2.9246 ...
    3.0833    1.5910    1.5266    1.6256    2.5037    1.4384];

Z = [
    1.7451    2.6345    0.5937   -2.8217    3.0304    1.0917   -1.7793 ...
    1.2422    2.1873   -2.3008   -3.3258    2.7617    0.9166    0.0601 ...
   -2.6520   -3.3205    4.1229   -3.4085   -3.1594   -0.7311; ...
   -3.2010   -4.9921   -3.7621   -4.7420   -4.1315   -3.9120   -4.5596 ...
   -4.9499   -3.4310   -4.2656   -6.2023   -4.5186   -3.7659   -5.0039 ...
   -4.3744   -5.0559   -3.9443   -4.0412   -5.3493   -3.0465];

cvx_begin
    variables a1(2) a2(2) a3(2) b1 b2 b3 
    a1'*X - b1 - ( a2'*X - b2 ) >= 1
    a1'*X - b1 - ( a3'*X - b3 ) >= 1
    a2'*Y - b2 - ( a1'*Y - b1 ) >= 1
    a2'*Y - b2 - ( a3'*Y - b3 ) >= 1
    a3'*Z - b3 - ( a1'*Z - b1 ) >= 1
    a3'*Z - b3 - ( a2'*Z - b2 ) >= 1  
    a1 + a2 + a3 == 1
    b1 + b2 + b3 == 1
cvx_end

% comment out the following line after filling in cvx part!
% values below are not right!!
%a1=[1;1];a2=[1;-5];a3=[-1;-1];b1=0;b2=0;b3=0;

% now let's plot the three-way separation induced by
% a1,a2,a3,b1,b2,b3
% find maximally confusing point
p = [(a1-a2)';(a1-a3)']\[(b1-b2);(b1-b3)];

% plot 
t = [-7:0.01:7];
u1 = a1-a2; u2 = a2-a3; u3 = a3-a1;
v1 = b1-b2; v2 = b2-b3; v3 = b3-b1;
line1 = (-t*u1(1)+v1)/u1(2); idx1 = find(u2'*[t;line1]-v2>0);
line2 = (-t*u2(1)+v2)/u2(2); idx2 = find(u3'*[t;line2]-v3>0);
line3 = (-t*u3(1)+v3)/u3(2); idx3 = find(u1'*[t;line3]-v1>0);

plot(X(1,:),X(2,:),'*',Y(1,:),Y(2,:),'ro',Z(1,:),Z(2,:),'g+',...
     t(idx1),line1(idx1),'k',t(idx2),line2(idx2),'k',t(idx3),line3(idx3),'k');
axis([-7 7 -7 7]);

Result

>> [a1 a2 a3]

ans =

    7.2480   -4.7801   -1.4680
    3.3939    4.9665   -7.3604

>> [b1 b2 b3]

ans =

    3.1130   -8.0986    5.9856