Pullback metric "Hello world"

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%%Paste commands into MATLAB window or download script

clear all; clc; close all;

%% Consider the set of points on the real line

x = -3e1:0.1:3e1;

y = zeros(length(x)); %just for plotting in 2d

%% Lets have a look

figure(1)

plot(x,y,'rx');

hold on

plot(x(1),y(1),'bo','Linewidth',5);

hold on

plot(x(end),y(end),'go','Linewidth',5);

%print(gcf, '-dpdf', '-r600', 'real line')

%% The euclidean distance metric `d' between two points in the set is |x1-x2|

func_d_normal = @(x1,x2)(d(x1,x2));

%% lets see, the distance between -30 and 30 should be 60..

test_d_euclidean = func_d_normal(x(1),x(end))

%% Now define a bijective function `f' which maps R to R^2,

%% and a `pullback' distance metric as follows:

func_d_pullback = @(x1,x2)(d2( f(x1), f(x2) ));

%% lets test it out

test_d2_pullback = func_d_pullback(x(1),x(end))

%% Lets have a look at the bijective mapping

x_new = f(x');

figure(2)

plot(x_new(:,1),x_new(:,2),'ro');

hold on

plot(x_new(1,1),x_new(1,2),'bo','Linewidth',5);

hold on

plot(x_new(end,1),x_new(end,2),'go','Linewidth',5);

print(gcf, '-dpdf', '-r600', 'mapped line')

%% Plot pairwise distance matrix between point in the set `x'

figure(3)

D = pdist2(x',x',func_d_normal); contourf(x,x,D,20); axis equal, colorbar; axis([min(x) max(x) min(x) max(x)])

hold on, plot([min(x) max(x)],[0 0],'k')

hold on, plot([0 0 ],[min(x) max(x)],'k')

%print(gcf, '-dpdf', '-r600', 'dist_matrix_euclidean')

%% Plot pairwise `pullback' distance matrix between point in the set `x'

figure(4)

D2 = pdist2(x',x',func_d_pullback); contourf(x,x,D2,8); axis equal, colorbar; axis([min(x) max(x) min(x) max(x)])

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