Show Sidebar Hide Sidebar

# Manifold Learning on Handwritten Digits in Scikit-learn

An illustration of various embeddings on the digits dataset.

The RandomTreesEmbedding, from the sklearn.ensemble module, is not technically a manifold embedding method, as it learn a high-dimensional representation on which we apply a dimensionality reduction method. However, it is often useful to cast a dataset into a representation in which the classes are linearly-separable.

t-SNE will be initialized with the embedding that is generated by PCA in this example, which is not the default setting. It ensures global stability of the embedding, i.e., the embedding does not depend on random initialization.

#### New to Plotly?¶

You can set up Plotly to work in online or offline mode, or in jupyter notebooks.
We also have a quick-reference cheatsheet (new!) to help you get started!

### Version¶

In [1]:
import sklearn
sklearn.__version__

Out[1]:
'0.18.1'

### Imports¶

In [2]:
import plotly.plotly as py
import plotly.graph_objs as go

from time import time

import numpy as np
import matplotlib.pyplot as plt
from sklearn import (manifold, datasets, decomposition, ensemble,
discriminant_analysis, random_projection)


### Calculations¶

In [3]:
digits = datasets.load_digits(n_class=6)
X = digits.data
y = digits.target
n_samples, n_features = X.shape
n_neighbors = 30


### Plot Results¶

In [4]:
def plot_embedding(X, title=None):
x_min, x_max = np.min(X, 0), np.max(X, 0)
X = (X - x_min) / (x_max - x_min)
anno=[]

for i in range(X.shape[0]):
anno.append(dict(x=X[i, 0], y=X[i, 1], text=str(digits.target[i]),
showarrow=False,
font=dict(
size=11,
color='rgb'+str(plt.cm.Set1(y[i] / 10.)[:3])),
))

shown_images = np.array([[1., 1.]])  # just something big

for i in range(digits.data.shape[0]):
dist = np.sum((X[i] - shown_images) ** 2, 1)
if np.min(dist) < 4e-3:
# don't show points that are too close
continue
shown_images = np.r_[shown_images, [X[i]]]

x_ = []
y_ = []
data = []
for i in range(0, len(shown_images)):
x_.append(shown_images[i][0])
y_.append(shown_images[i][1])

data.append(shown_images)

trace = go.Scatter(x=x_, y=y_,
showlegend=False,
mode='markers',
marker=dict(color='white', size=15,
line=dict(color='black', width=1)))
layout = go.Layout(annotations=anno, title=title,
xaxis=dict(ticks='', showticklabels=False,
showgrid=False, zeroline=False),
yaxis=dict(ticks='', showticklabels=False,
showgrid=False, zeroline=False),
)
fig = go.Figure(data=[trace], layout=layout)

return fig


### Plot Images of the Digits¶

In [5]:
n_img_per_row = 20
img = np.zeros((10 * n_img_per_row, 10 * n_img_per_row))
for i in range(n_img_per_row):
ix = 10 * i + 1
for j in range(n_img_per_row):
iy = 10 * j + 1
img[ix:ix + 8, iy:iy + 8] = X[i * n_img_per_row + j].reshape((8, 8))

In [6]:
def matplotlib_to_plotly(cmap, pl_entries):
h = 1.0/(pl_entries-1)
pl_colorscale = []

for k in range(pl_entries):
C = map(np.uint8, np.array(cmap(k*h)[:3])*255)
pl_colorscale.append([k*h, 'rgb'+str((C[0], C[1], C[2]))])

return pl_colorscale

trace = go.Heatmap(z=img,
colorscale=matplotlib_to_plotly(plt.cm.binary, 5),
showscale=False)

layout = go.Layout(title='A selection from the 64-dimensional digits dataset',
xaxis=dict(ticks='', showticklabels=False),
yaxis=dict(ticks='', showticklabels=False,
autorange='reversed'),
)
fig = go.Figure(data=[trace], layout=layout)

In [7]:
py.iplot(fig)

Out[7]:

### Random 2D projection using a random unitary matrix¶

In [8]:
print("Computing random projection")
rp = random_projection.SparseRandomProjection(n_components=2, random_state=42)
X_projected = rp.fit_transform(X)
fig = plot_embedding(X_projected, "Random Projection of the digits")

Computing random projection

In [9]:
py.iplot(fig)

Out[9]:

### Projection on to the first 2 principal components¶

In [10]:
print("Computing PCA projection")
t0 = time()
X_pca = decomposition.TruncatedSVD(n_components=2).fit_transform(X)

fig = plot_embedding(X_pca,
"Principal Components projection of the digits (time %.2fs)" %
(time() - t0))

Computing PCA projection

In [11]:
py.iplot(fig)

Out[11]:

### Projection on to the first 2 linear discriminant components¶

In [12]:
print("Computing Linear Discriminant Analysis projection")
X2 = X.copy()
X2.flat[::X.shape[1] + 1] += 0.01  # Make X invertible
t0 = time()
X_lda = discriminant_analysis.LinearDiscriminantAnalysis(n_components=2).fit_transform(X2, y)

fig = plot_embedding(X_lda,
"Linear Discriminant projection of the digits (time %.2fs)" %
(time() - t0))

Computing Linear Discriminant Analysis projection

In [13]:
py.iplot(fig)

Out[13]:

### Isomap projection of the digits dataset¶

In [14]:
print("Computing Isomap embedding")
t0 = time()
X_iso = manifold.Isomap(n_neighbors, n_components=2).fit_transform(X)
print("Done.")

fig = plot_embedding(X_iso,
"Isomap projection of the digits (time %.2fs)" %
(time() - t0))

Computing Isomap embedding
Done.

In [15]:
py.iplot(fig)

Out[15]:

### Locally linear embedding of the digits dataset¶

In [16]:
print("Computing LLE embedding")
clf = manifold.LocallyLinearEmbedding(n_neighbors, n_components=2,
method='standard')
t0 = time()
X_lle = clf.fit_transform(X)
print("Done. Reconstruction error: %g" % clf.reconstruction_error_)

fig = plot_embedding(X_lle,
"Locally Linear Embedding of the digits (time %.2fs)" %
(time() - t0))

Computing LLE embedding
Done. Reconstruction error: 1.63544e-06

In [17]:
py.iplot(fig)

Out[17]:

### Modified Locally linear embedding of the digits dataset¶

In [18]:
print("Computing modified LLE embedding")
clf = manifold.LocallyLinearEmbedding(n_neighbors, n_components=2,
method='modified')
t0 = time()
X_mlle = clf.fit_transform(X)
print("Done. Reconstruction error: %g" % clf.reconstruction_error_)

fig = plot_embedding(X_mlle,
"Modified Locally Linear Embedding of the digits (time %.2fs)" %
(time() - t0))

Computing modified LLE embedding
Done. Reconstruction error: 0.360668

In [19]:
py.iplot(fig)

Out[19]:

### HLLE embedding of the digits dataset¶

In [20]:
print("Computing Hessian LLE embedding")
clf = manifold.LocallyLinearEmbedding(n_neighbors, n_components=2,
method='hessian')
t0 = time()
X_hlle = clf.fit_transform(X)
print("Done. Reconstruction error: %g" % clf.reconstruction_error_)

fig = plot_embedding(X_hlle,
"Hessian Locally Linear Embedding of the digits (time %.2fs)" %
(time() - t0))

Computing Hessian LLE embedding
Done. Reconstruction error: 0.212801

In [21]:
py.iplot(fig)

Out[21]:

### LTSA embedding of the digits dataset¶

In [22]:
print("Computing LTSA embedding")
clf = manifold.LocallyLinearEmbedding(n_neighbors, n_components=2,
method='ltsa')
t0 = time()
X_ltsa = clf.fit_transform(X)
print("Done. Reconstruction error: %g" % clf.reconstruction_error_)
fig = plot_embedding(X_ltsa,
"Local Tangent Space Alignment of the digits (time %.2fs)" %
(time() - t0))

Computing LTSA embedding
Done. Reconstruction error: 0.212804

In [23]:
py.iplot(fig)

Out[23]:

### MDS embedding of the digits dataset¶

In [24]:
print("Computing MDS embedding")
clf = manifold.MDS(n_components=2, n_init=1, max_iter=100)
t0 = time()
X_mds = clf.fit_transform(X)
print("Done. Stress: %f" % clf.stress_)
fig = plot_embedding(X_mds,
"MDS embedding of the digits (time %.2fs)" %
(time() - t0))

Computing MDS embedding
Done. Stress: 143118271.858794

In [25]:
py.iplot(fig)

Out[25]:

### Random Trees embedding of the digits dataset¶

In [26]:
print("Computing Totally Random Trees embedding")
hasher = ensemble.RandomTreesEmbedding(n_estimators=200, random_state=0,
max_depth=5)
t0 = time()
X_transformed = hasher.fit_transform(X)
pca = decomposition.TruncatedSVD(n_components=2)
X_reduced = pca.fit_transform(X_transformed)

fig = plot_embedding(X_reduced,
"Random forest embedding of the digits (time %.2fs)" %
(time() - t0))

Computing Totally Random Trees embedding

In [27]:
py.iplot(fig)

Out[27]:

### Spectral embedding of the digits dataset¶

In [28]:
print("Computing Spectral embedding")
embedder = manifold.SpectralEmbedding(n_components=2, random_state=0,
eigen_solver="arpack")
t0 = time()
X_se = embedder.fit_transform(X)

fig = plot_embedding(X_se,
"Spectral embedding of the digits (time %.2fs)" %
(time() - t0))

Computing Spectral embedding

In [29]:
py.iplot(fig)

Out[29]:

### t-SNE embedding of the digits dataset¶

In [30]:
print("Computing t-SNE embedding")
tsne = manifold.TSNE(n_components=2, init='pca', random_state=0)
t0 = time()
X_tsne = tsne.fit_transform(X)

fig = plot_embedding(X_tsne,
"t-SNE embedding of the digits (time %.2fs)" %
(time() - t0))

Computing t-SNE embedding

In [31]:
py.iplot(fig)

Out[31]:

Authors:

      Fabian Pedregosa <fabian.pedregosa@inria.fr>

Olivier Grisel <olivier.grisel@ensta.org>

Mathieu Blondel <mathieu@mblondel.org>

Gael Varoquaux



      BSD 3 clause (C) INRIA 2011