Comparison of Dimension Reduction Techniques¶

A comparison of several different dimension reduction techniques on a variety of toy datasets. The datasets are all toy datasets, but should provide a representative range of the strengths and weaknesses of the different algorithms.

The time to perform the dimension reduction with each algorithm and each dataset is recorded in the lower right of each plot.

Things to note about the datasets:

• Blobs: A set of five gaussian blobs in 10 dimensional
space. This should be a prototypical example of something that should clearly separate even in a reduced dimension space.
• Iris: a classic small dataset with one distinct class
and two classes that are not clearly separated.
• Digits: handwritten digits – ideally different digit
classes should form distinct groups. Due to the nature of handwriting digits may have several forms (crossed or uncrossed sevens, capped or straight line oes, etc.)
• Wine: wine characteristics ideally used for a toy
regression. Ultimately the data is essentially one dimensional in nature.
• Swiss Roll: data is essentially a rectangle, but
has been “rolled up” like a swiss roll in three dimensional space. Ideally a dimension reduction technique should be able to “unroll” it. The data has been coloured according to one dimension of the rectangle, so should form a rectangle of smooth color variation.
• Sphere: the two dimensional surface of a three
dimensional sphere. This cannot be represented accurately in two dimensions without tearing. The sphere has been coloured with hue around the equator and black to white from the south to north pole.
import numpy as np
import matplotlib.pyplot as plt
import seaborn as sns
import time

from sklearn import datasets, decomposition, manifold, preprocessing
from colorsys import hsv_to_rgb

import umap

sns.set(context="paper", style="white")

blobs, blob_labels = datasets.make_blobs(
n_samples=500, n_features=10, centers=5, random_state=42
)
swissroll, swissroll_labels = datasets.make_swiss_roll(
n_samples=1000, noise=0.1, random_state=42
)
sphere = np.random.normal(size=(600, 3))
sphere = preprocessing.normalize(sphere)
sphere_hsv = np.array(
[
(
(np.arctan2(c[1], c[0]) + np.pi) / (2 * np.pi),
np.abs(c[2]),
min((c[2] + 1.1), 1.0),
)
for c in sphere
]
)
sphere_colors = np.array([hsv_to_rgb(*c) for c in sphere_hsv])

reducers = [
(manifold.TSNE, {"perplexity": 50}),
# (manifold.LocallyLinearEmbedding, {'n_neighbors':10, 'method':'hessian'}),
(manifold.Isomap, {"n_neighbors": 30}),
(manifold.MDS, {}),
(decomposition.PCA, {}),
(umap.UMAP, {"n_neighbors": 30, "min_dist": 0.3}),
]

test_data = [
(blobs, blob_labels),
(iris.data, iris.target),
(digits.data, digits.target),
(wine.data, wine.target),
(swissroll, swissroll_labels),
(sphere, sphere_colors),
]
dataset_names = ["Blobs", "Iris", "Digits", "Wine", "Swiss Roll", "Sphere"]

n_rows = len(test_data)
n_cols = len(reducers)
ax_index = 1
ax_list = []

# plt.figure(figsize=(9 * 2 + 3, 12.5))
plt.figure(figsize=(10, 8))
left=.02, right=.98, bottom=.001, top=.96, wspace=.05, hspace=.01
)
for data, labels in test_data:
for reducer, args in reducers:
start_time = time.time()
embedding = reducer(n_components=2, **args).fit_transform(data)
elapsed_time = time.time() - start_time
ax = plt.subplot(n_rows, n_cols, ax_index)
if isinstance(labels[0], tuple):
ax.scatter(*embedding.T, s=10, c=labels, alpha=0.5)
else:
ax.scatter(
*embedding.T, s=10, c=labels, cmap="Spectral", alpha=0.5
)
ax.text(
0.99,
0.01,
"{:.2f} s".format(elapsed_time),
transform=ax.transAxes,
size=14,
horizontalalignment="right",
)
ax_list.append(ax)
ax_index += 1
plt.setp(ax_list, xticks=[], yticks=[])

for i in np.arange(n_rows) * n_cols:
ax_list[i].set_ylabel(dataset_names[i // n_cols], size=16)
for i in range(n_cols):
ax_list[i].set_xlabel(repr(reducers[i][0]()).split("(")[0], size=16)
ax_list[i].xaxis.set_label_position("top")

plt.tight_layout()
plt.show()


Total running time of the script: ( 0 minutes 0.000 seconds)

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