Manifold Learning with Isomap¶

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This tour explores the Isomap algorithm for manifold learning.

The <http://waldron.stanford.edu/~isomap/ Isomap> algorithm is introduced in

A Global Geometric Framework for Nonlinear Dimensionality Reduction, J. B. Tenenbaum, V. de Silva and J. C. Langford, Science 290 (5500): 2319-2323, 22 December 2000.

In [1]:

using PyPlot
using NtToolBox

Graph Approximation of Manifolds¶

Manifold learning consist in approximating the parameterization of a manifold represented as a point cloud.

First we load a simple 3D point cloud, the famous Swiss Roll.

Number of points.

In [2]:

n = 500; #1000 points leads to very slow computation

Random position on the parameteric domain.

In [3]:

x = rand(2,n);

Mapping on the manifold.

In [4]:

v = 3*pi/2*(.1 + 2*x[1,:])
X  = zeros(3,n)
X[2,:] = 20*x[2,:]
X[1,:] = - cos(v).*v
X[3,:] = sin(v).*v;

Parameter for display.

In [5]:

ms = 200
elev = 20; azim = -110;

Display the point cloud.

In [6]:

fig = figure(figsize=(15,11))
ax = gca(projection="3d")

#swiss roll
scatter3D(X[1,:], X[2,:], X[3,:], c=get_cmap("jet")((X[1,:].^2+X[3,:].^2)/100), s=ms, lw=0, alpha=1)

#params
xlim(minimum(X[1,:]),maximum(X[1,:]))
ylim(minimum(X[2,:]),maximum(X[2,:]))
zlim(minimum(X[3,:]),maximum(X[3,:]))
axis("off")
ax[:view_init](elev, azim)

Compute the pairwise Euclidean distance matrix.

In [7]:

D1 = repeat(sum(X.^2, 1), outer=(n,1))
D1 = D1 + D1' - 2*X'*X
D1[D1.<0] = NaN
D1=sqrt(D1);

Number of NN for the graph.

In [8]:

k = 6;

Compute the k-NN connectivity.

In [9]:

DNN, NN = zeros(size(D1)), zeros(size(D1))
for i in 1:size(D1,2)
    DNN[i,:], NN[i,:] = sort(D1[i,:]), sortperm(D1[i,:])
end
NN = Int.(NN[:,2:k+1])
DNN = DNN[:,2:k+1];

Adjacency matrix, and weighted adjacency.

In [10]:

B = repeat((1:n)', outer=(k,1))
A = sparse(vec(B), vec(NN'), ones(k*n));

Weighted adjacency (the metric on the graph).

In [11]:

W = sparse(vec(B),vec(NN'), vec(DNN'));

Display the graph.

In [12]:

fig = figure(figsize=(15,11))
ax = gca(projection="3d")

#swiss roll
scatter3D(X[1,:], X[2,:], X[3,:], c=get_cmap("jet")((X[1,:].^2+X[3,:].^2)/100), s=ms, lw=0, alpha=1)

#graph
(I,J) = findn(A)
#(I,J) = (vec(B), vec(NN))
xx = hcat(X[1,I],X[1,J])
yy = hcat(X[2,I],X[2,J])
zz = hcat(X[3,I],X[3,J])

for i in 1:length(I)
    plot3D(xx[i,:], yy[i,:], zz[i,:], color="black")
end
    
#params
xlim(minimum(X[1,:]),maximum(X[1,:]))
ylim(minimum(X[2,:]),maximum(X[2,:]))
zlim(minimum(X[3,:]),maximum(X[3,:]))
axis("off")
ax[:view_init](elev, azim)