%%

U = [1 -1; 1 1]/sqrt(2);
S = [1 0 ; 0 .01];
V = normc([1 1; 2 -1; -1 3; -2 2]);
X = U*S*V';

figure(2);clf;scatter(X(1,:),X(2,:),50,'filled');axis square
colorbar

[U2,S2,V2] = svd(X)

%%


p = 2;
n = 100;
u_orig = [1 -1; 1 1]/sqrt(2);
v_orig = randn(p,n)/sqrt(n);
X = u_orig*[1 0; 0 1e-10]*v_orig;
w = rand(2,1)
noise = randn(n,1)/100;
y = X'*w + noise;%v_orig'*[1;.1];

figure(1);clf;scatter(X(1,:),X(2,:),50,y,'filled');axis square
colorbar

[U,S,V] = svds(X,p);

%LS
what_LS = U*inv(S)*V'*y

%Tik
lambda = .01;
S2 = diag(diag(S)./(diag(S).^2 + lambda));
what_tik = U*S2*V'*y

%%

p = 2;
n = 100;
U = orth(randn(p))
Sigma = diag([1 .1]);
V = randn(n,p);

X = U*Sigma*V';

figure(4);clf;scatter(X(1,:),X(2,:),50,'filled');axis square
axis([-1,1,-1,1])
colorbar
grid on
hold on
for k = 1:2
   quiver(0,0,U(1,k)*Sigma(k,k),U(2,k)*Sigma(k,k),0,'filled','linewidth',5,'maxheadsize',.5)
   %    quiver(0,0,U(1,k),U(2,k),0,'filled','linewidth',5,'maxheadsize',.5)
end