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


2. Regression:

%% Interpolation in 1-D

% Interpolation set
n=10; x=1:n; % 0,1,2,...,n
y=sin(x);  % sparse samples of a smooth function.

% Plot the (default) piecewise linear interpolation
plot(x,y);  % plot 1: "connect the dots" by default

% Polynomial regression with Vandermonde matrix:
p=vander(x)\y';  % coefficients of the interpolating polynomial
figure; z=linspace(0,n,200); plot(z,polyval(p,z)); % plot 2

% Alternatively, use the built-in Octave function:
p=polyfit(x,y,n);
figure; plot(z,polyval(p,z)); % plot 3: polynomial of degree n

% Spline interpolation:
pp=spline(x,y);
figure; plot(z,ppval(pp,z)); % plot 4: piecewise cubic spline

% Piecewise linear interpolation usable for computation:
pl=interp1(x,y,"linear","pp");
figure; plot(z,ppval(pl,z)); % plot 5: piecewise linear polynomial

%% Interpolation in 2-D

% Square lattice of points in the plane:
t=-9:3:9; [xx,yy]=meshgrid(t,t); zz=xx.^2+yy.^2;
% Piecewise linear graph plots
figure; surf(xx,yy,zz);
figure; mesh(xx,yy,zz);
% Piecewise linear level curves through many linearly interpolated levels
figure; contour(xx,yy,zz);
% Sample more often to get smaller errors from smooth:
t=-9:0.5:9; [xx,yy]=meshgrid(t,t); zz=xx.^2+yy.^2;
figure; surf(xx,yy,zz); figure; mesh(xx,yy,zz); figure; contour(xx,yy,zz);


%% Convex hulls and triangulations
% 2-simplex in the plane
x=[0,0,1]; y=[0,1,0]; % triangle (0,0), (0,1), (1,0)
k=convhull(x,y); plot(x(k),y(k));
% Random points in [0,1]x[0,1]
xy=rand(17,2); k=convhull(xy); figure; plot(xy(k,1),xy(k,2));

% 3-simplex in R^3
x=[0,0,0,1]; y=[0,0,1,0]; z=[0,1,0,0];
k=convhull([x',y',z'])  % 4 faces of the tetrahedron, 1 per row
figure; trimesh(k,x,y,z);  % plot as a wireframe, or mesh, in 3D
figure; trisurf(k,x,y,z);  % plot using colored surfaces in 3D

% Random points in R^4
N=20; xyzw=rand(N,4);
[h,vol]=convhulln(xyzw); h  % listing of points in each 3-simplex face
vol % volume of the convex hull
% project the hull into R^3 along the 4 coordinates axes:
figure; trimesh(h,xyzw(:,1),xyzw(:,2),xyzw(:,3))   % eliminate axis 4
figure; trimesh(h,xyzw(:,1),xyzw(:,2),xyzw(:,4))   % eliminate axis 3
figure; trimesh(h,xyzw(:,1),xyzw(:,3),xyzw(:,4))   % eliminate axis 2
figure; trimesh(h,xyzw(:,2),xyzw(:,3),xyzw(:,4))   % eliminate axis 1

% Plot a Delaunay triangulation 2D
x=rand(1,20); y=rand(1,20);  % 20 random points in [0,1]x[0,1]
i=delaunay(x,y);  % indices in {(x,y)} of the vertices of Delaunay triangles
figure; triplot(i,x,y);  % plot the triangles

% Delaunay triangulation of a paraboloid in 3D from parameter space
x=2*rand(1,20)-1; y=2*rand(1,20)-1;  % 20 random points in parameter space [-1,1]x[-1,1]
i=delaunay(x,y);  % indices of the Delaunay triangles
figure; triplot(i,x,y);  % plot the triangles in the parameter space
z=x.^2+y.^2;  % lift onto the paraboloid using the parametrization.
figure; trimesh(i,x,y,z);  % display the triangulated manifold in 3D as a mesh 
figure; trisurf(i,x,y,z);  % display the triangulated manifold in 3D as surfaces

% Delaunay tesselation in 3D
x=rand(1,10); y=rand(1,10); z=rand(1,10) % 10 random points in [0,1]x[0,1]x[0,1]
j=delaunay(x,y,z);
figure; tetramesh(j,x,y,z);  % plot the component 3-simplexes as solids of one color