DIMENSIONALITY REDUCTION AND MANIFOLD ESTIMATION
Mlden Victor Wickerhauser
PMF/Matematika -- University of Zagreb
Winter, 2022
Example Computer Programs (for Octave/Matlab)


1. Manifolds:

%% Piecewise linear plots of the unit 2-sphere with 10, 20, and 50 latitudes 
for n=[10,20,50]
 [x,y,z]=sphere(n); figure; surf(x,y,z); axis equal;
end


%% Graph parametrizations of the unit 2-sphere embedded in E^3.

F  = @(p) sumsq(p)-1;   % M={p:F(p)=0}
DF = @(p) 2*p; % maximal rank 1 at all p in M

p=[0,0,-1]; % south pole
F(p)        % should be 0, indicating that p is on M
rank(DF(p)) % should be 1, the maximal rank
t=-0.5:0.05:0.5; [xx,yy]=meshgrid(t,t);  % parameter space (xx,yy) near (0,0)
zz= -sqrt(1-xx.^2-yy.^2);   % implicit function giving (xx,yy,zz) on M
figure; mesh(xx,yy,zz); axis equal;  % plot of the graph


%% Union of two unit 2-spheres embedded in R^4 as a differentiable variety

FF = @(p) [sumsq(p)-2; p(4)^2-1];
DFF = @(p) [2*p; 0,0,0,2*p(4)];

p=[0,0,1,1]; % where is this? the north pole of the upper 2-sphere!
FF(p)        % should be [0;0], indicating that p is on M
rank(DFF(p)) % should be 2, the maximal rank
t=-0.5:0.05:0.5; [xx,yy]=meshgrid(t,t);  % parameter space (xx,yy) near (0,0)
zz= sqrt(1-xx.^2-yy.^2);   % implicit function giving (xx,yy,zz,1) on M
figure; mesh(xx,yy,zz); axis equal;  % plot of the graph