The Isomap algorithm is an approach to manifold learning. Isomap seeks a lower dimensional embedding of a set of high dimensional data points estimating the intrinsic geometry of a data manifold based on a rough estimate of each data point’s neighbors.
The scikit-learn library provides a great implmentation of the Isomap algorithm and a dataset of handwritten digits. In this post we'll see how to load the dataset and how to compute an embedding of the dataset on a bidimentional space.
Let's load the dataset and show some samples

from pylab import scatter,text,show,cm,figure
from pylab import subplot,imshow,NullLocator
from sklearn import manifold, datasets

# load the digits dataset
# about 180 samples for the digits (0,1,2,3,4)
digits = datasets.load_digits(n_class=5)
X = digits.data
color = digits.target

# shows some digits
figure(1)
for i in range(36):
 ax = subplot(6,6,i)
 ax.xaxis.set_major_locator(NullLocator()) # remove ticks
 ax.yaxis.set_major_locator(NullLocator())
 imshow(digits.images[i], cmap=cm.gray_r)

The result should be as follows:

Now X is a matrix where each row is a vector that represent a digit. Each vector has 64 elements and it has been obtained using spatial resampling on the above images. We can apply the Isomap algorithm on this data and plot the result with the following lines:

# running Isomap
# 5 neighbours will be considered and reduction on a 2d space
Y = manifold.Isomap(5, 2).fit_transform(X)

# plotting the result
figure(2)
scatter(Y[:,0], Y[:,1], c='k', alpha=0.3, s=10)
for i in range(Y.shape[0]):
 text(Y[i, 0], Y[i, 1], str(color[i]),
      color=cm.Dark2(color[i] / 5.),
      fontdict={'weight': 'bold', 'size': 11})
show()

The new embedding for the data will be as follows: