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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.
from __future__ import division
import numpy as np
import scipy as scp
import pylab as pyl
import matplotlib.pyplot as plt
from nt_toolbox.general import *
from nt_toolbox.signal import *
import warnings
warnings.filterwarnings('ignore')
%matplotlib inline
%load_ext autoreload
%autoreload 2
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.
n = 1000
Random position on the parameteric domain.
from numpy import random
x = random.rand(2,n)
Mapping on the manifold.
v = 3*np.pi/2*(.1 + 2*x[0,:])
X = np.zeros([3,n])
X[1,:] = 20*x[1,:]
X[0,:] = - np.cos(v)*v
X[2,:] = np.sin(v)*v
Parameter for display.
ms = 200
el = 20; az = -110
Display the point cloud.
from mpl_toolkits.mplot3d import Axes3D
fig = plt.figure(figsize=(15,11))
ax = fig.add_subplot(111, projection="3d")
#swiss roll
ax.scatter(X[0,:], X[1,:], X[2,:], c=plt.cm.jet((X[0,:]**2+X[2,:]**2)/100), s=ms, lw=0, alpha=1)
#params
ax.set_xlim(np.min(X[0,:]),np.max(X[0,:]))
ax.set_ylim(np.min(X[1,:]),np.max(X[1,:]))
ax.set_zlim(np.min(X[2,:]),np.max(X[2,:]))
ax.axis("off")
ax.view_init(elev=el, azim=az)
Compute the pairwise Euclidean distance matrix.
D1 = np.repeat(np.sum(X**2, 0)[:,np.newaxis], n, 1)
D1 = np.sqrt(D1 + np.transpose(D1) - 2*np.dot(np.transpose(X), X))
Number of NN for the graph.
k = 6
Compute the k-NN connectivity.
DNN, NN = np.sort(D1), np.argsort(D1)
NN = NN[:,1:k+1]
DNN = DNN[:,1:k+1]
Adjacency matrix, and weighted adjacency.
from scipy import sparse
B = np.tile(np.arange(0,n),(k,1))
A = sparse.coo_matrix((np.ones(k*n),(np.ravel(B, order="F"), np.ravel(NN))))
Weighted adjacency (the metric on the graph).
W = sparse.coo_matrix((np.ravel(DNN),(np.ravel(B, order="F"), np.ravel(NN))))
Display the graph.
from mpl_toolkits.mplot3d import Axes3D
fig = plt.figure(figsize=(15,11))
ax = fig.add_subplot(111, projection="3d")
#swiss roll
ax.scatter(X[0,:], X[1,:], X[2,:], c=plt.cm.jet((X[0,:]**2+X[2,:]**2)/100), s=ms, lw=0, alpha=1)
#graph
I,J,V = sparse.find(A)
xx = np.vstack((X[0,I],X[0,J]))
yy = np.vstack((X[1,I],X[1,J]))
zz = np.vstack((X[2,I],X[2,J]))
for i in range(len(I)):
ax.plot(xx[:,i], yy[:,i], zz[:,i], color="black")
#params
ax.axis("off")
ax.set_xlim(np.min(X[0,:]),np.max(X[0,:]))
ax.set_ylim(np.min(X[1,:]),np.max(X[1,:]))
ax.set_zlim(np.min(X[2,:]),np.max(X[2,:]))
ax.view_init(elev=el, azim=az)
plt.show()
A simple algorithm to compute the geodesic distances between all pairs of points on a graph is Floyd iterative algorithm. Its complexity is $\mathcal O(n^3)$ where $n$ is the number of points. It is thus quite slow for sparse graph, where Dijkstra runs in $\mathcal O(n^2\log(n))$.
Floyd algorithm iterates the following update rule, for $k=1,\dots,n$
$D(i,j) \leftarrow \min(D(i,j), D(i,k)+D(k,j))$,
with the initialization $D(i,j)=W(i,j)$ if $W(i,j)>0$, and $D(i,j)=Inf$ if $W(i,j)=0$.
Make the graph symmetric.
D = W.toarray()
D = (D + np.transpose(D))/2.
Initialize the matrix.
D[D == 0] = np.float("inf")
Add connexion between a point and itself.
D = D - np.diag(np.diag(D))
D[np.isnan(D)] = np.float("inf")
Exercise 1
Implement the Floyd algorithm to compute the full distance matrix $D$, where $D(i,j)$ is the geodesic distance between
run -i nt_solutions/shapes_7_isomap/exo1
## Insert your code here.
Find index of vertices that are not connected to the main manifold.
Iremove = np.where(D[:,0] == np.float("Inf"))
Remove Inf remaining values (disconnected components).
D[D == np.float("Inf")] = 0
Isomap perform the dimensionality reduction by applying multidimensional scaling.
Please refers to the tours on Bending Invariant for detail on Classical MDS (strain minimization).
Exercise 2
Perform classical MDS to compute the 2D flattening.
run -i nt_solutions/shapes_7_isomap/exo2
## Insert your code here.
Redess the points using the two leading eigenvectors of the covariance matrix (PCA correction).
[L, U] = linalg.eig(np.dot(Xstrain, np.transpose(Xstrain))/n)
Xstrain1 = np.dot(np.transpose(U), Xstrain)
Remove problematic points.
Xstrain1[:,Iremove] = np.float("inf")
Display the final result of the dimensionality reduction.
#plot size
plt.figure(figsize = (15,6))
#plot points
plt.scatter(Xstrain1[0,:], Xstrain1[1,:], ms, c=plt.cm.jet((X[0,:]**2+X[2,:]**2)/100), lw=0, alpha=1)
#plot vertices
I,J,V = sparse.find(A)
xx = np.vstack((Xstrain1[0,I], Xstrain1[0,J]))
yy = np.vstack((Xstrain1[1,I], Xstrain1[1,J]))
for i in range(len(I)):
plt.plot(xx[:,i], yy[:,i], color="black")
#params
plt.axis("off")
plt.xlim(np.min(Xstrain1[0,:]-1),np.max(Xstrain1[0,:])+1)
plt.ylim(np.min(Xstrain1[1,:]-1),np.max(Xstrain1[1,:])+1)
plt.show()
For comparison, the ideal locations on the parameter domain.
Y = np.vstack((v, X[1,:]))
Y[0,:] = rescale(Y[0,:], min(Xstrain[0,:]), max(Xstrain[0,:]))
Y[1,:] = rescale(Y[1,:], min(Xstrain[1,:]), max(Xstrain[1,:]))
Display the ideal graph on the reduced parameter domain.
#plot size
plt.figure(figsize = (15,6))
#plot points
plt.scatter(Y[0,:], Y[1,:], ms, c=plt.cm.jet((X[0,:]**2+X[2,:]**2)/100), lw=0, alpha=1)
#plot vertices
I,J,V = sparse.find(A)
xx = np.vstack((Y[0,I], Y[0,J]))
yy = np.vstack((Y[1,I], Y[1,J]))
for i in range(len(I)):
plt.plot(xx[:,i], yy[:,i], color="black")
#params
plt.axis("off")
plt.xlim(np.min(Y[0,:]-1),np.max(Y[0,:])+1)
plt.ylim(np.min(Y[1,:]-1),np.max(Y[1,:])+1)
plt.show()
It is possible to use SMACOF instead of classical scaling.
Please refers to the tours on Bending Invariant for detail on both Classical MDS (strain minimization) and SMACOF MDS (stress minimization).
Exercise 3
Perform stress minimization MDS using SMACOF to compute the 2D flattening.
run -i nt_solutions/shapes_7_isomap/exo3
## Insert your code here.
Plot stress evolution during minimization.
plt.figure(figsize=(10,7))
plt.plot(stress, '.-')
plt.show()
Compute the main direction of the point clouds.
[L, U] = linalg.eig(np.dot(Xstress, np.transpose(Xstress))/n)
[L, I] = np.sort(L), np.argsort(L)
U = U[:,I[1:3]]
Project the points on the two leading eigenvectors of the covariance matrix (PCA final projection).
Xstress1 = np.dot(np.transpose(U), Xstress)
Remove problematic points.
Xstress1[:,Iremove] = np.float("Inf")
Display the final result of the dimensionality reduction.
#plot size
plt.figure(figsize = (15,6))
#plot points
plt.scatter(Xstress1[1,:], Xstress1[0,:], ms, c=plt.cm.jet((X[0,:]**2+X[2,:]**2)/100), lw=0, alpha=1)
#plot vertices
I,J,V = sparse.find(A)
xx = np.vstack((Xstress1[1,I], Xstress1[1,J]))
yy = np.vstack((Xstress1[0,I], Xstress1[0,J]))
for i in range(len(I)):
plt.plot(xx[:,i], yy[:,i], color="black")
#params
plt.axis("off")
plt.xlim(np.min(Xstress1[1,:]-1),np.max(Xstress1[1,:])+1)
plt.ylim(np.min(Xstress1[0,:]-1),np.max(Xstress1[0,:])+1)
plt.show()