截断球体上的流形学习#

不同流形学习技术在球形数据集上的应用。在这里，您可以看到降维的使用，以便对流形学习方法获得一些直觉。关于数据集，球体的两极被切除，其侧面也切除了一薄片。这使得流形学习技术能够“展开”它，同时将其投影到二维空间。

对于一个类似的示例，其中这些方法应用于 S 曲线数据集，请参见流形学习方法比较

请注意，MDS（多维缩放）的目的是找到数据的低维表示（此处为 2D），其中距离很好地遵循原始高维空间中的距离，与其他流形学习算法不同，它不寻求数据在低维空间中的各向同性表示。这里的流形问题相当类似于表示地球的平面地图，就像地图投影一样。

Manifold Learning with 1000 points, 10 neighbors, LLE (0.051 sec), LTSA (0.81 sec), Hessian LLE (0.71 sec), Modified LLE (1.2 sec), Isomap (0.19 sec), MDS (0.7 sec), Spectral Embedding (0.06 sec), t-SNE (3.7 sec)

standard: 0.051 sec
ltsa: 0.81 sec
hessian: 0.71 sec
modified: 1.2 sec
ISO: 0.19 sec
MDS: 0.7 sec
Spectral Embedding: 0.06 sec
t-SNE: 3.7 sec

# Authors: The scikit-learn developers
# SPDX-License-Identifier: BSD-3-Clause

from time import time

import matplotlib.pyplot as plt

# Unused but required import for doing 3d projections with matplotlib < 3.2
import mpl_toolkits.mplot3d  # noqa: F401
import numpy as np
from matplotlib.ticker import NullFormatter

from sklearn import manifold
from sklearn.utils import check_random_state

# Variables for manifold learning.
n_neighbors = 10
n_samples = 1000

# Create our sphere.
random_state = check_random_state(0)
p = random_state.rand(n_samples) * (2 * np.pi - 0.55)
t = random_state.rand(n_samples) * np.pi

# Sever the poles from the sphere.
indices = (t < (np.pi - (np.pi / 8))) & (t > ((np.pi / 8)))
colors = p[indices]
x, y, z = (
    np.sin(t[indices]) * np.cos(p[indices]),
    np.sin(t[indices]) * np.sin(p[indices]),
    np.cos(t[indices]),
)

# Plot our dataset.
fig = plt.figure(figsize=(15, 8))
plt.suptitle(
    "Manifold Learning with %i points, %i neighbors" % (1000, n_neighbors), fontsize=14
)

ax = fig.add_subplot(251, projection="3d")
ax.scatter(x, y, z, c=p[indices], cmap=plt.cm.rainbow)
ax.view_init(40, -10)

sphere_data = np.array([x, y, z]).T

# Perform Locally Linear Embedding Manifold learning
methods = ["standard", "ltsa", "hessian", "modified"]
labels = ["LLE", "LTSA", "Hessian LLE", "Modified LLE"]

for i, method in enumerate(methods):
    t0 = time()
    trans_data = (
        manifold.LocallyLinearEmbedding(
            n_neighbors=n_neighbors, n_components=2, method=method, random_state=42
        )
        .fit_transform(sphere_data)
        .T
    )
    t1 = time()
    print("%s: %.2g sec" % (methods[i], t1 - t0))

    ax = fig.add_subplot(252 + i)
    plt.scatter(trans_data[0], trans_data[1], c=colors, cmap=plt.cm.rainbow)
    plt.title("%s (%.2g sec)" % (labels[i], t1 - t0))
    ax.xaxis.set_major_formatter(NullFormatter())
    ax.yaxis.set_major_formatter(NullFormatter())
    plt.axis("tight")

# Perform Isomap Manifold learning.
t0 = time()
trans_data = (
    manifold.Isomap(n_neighbors=n_neighbors, n_components=2)
    .fit_transform(sphere_data)
    .T
)
t1 = time()
print("%s: %.2g sec" % ("ISO", t1 - t0))

ax = fig.add_subplot(257)
plt.scatter(trans_data[0], trans_data[1], c=colors, cmap=plt.cm.rainbow)
plt.title("%s (%.2g sec)" % ("Isomap", t1 - t0))
ax.xaxis.set_major_formatter(NullFormatter())
ax.yaxis.set_major_formatter(NullFormatter())
plt.axis("tight")

# Perform Multi-dimensional scaling.
t0 = time()
mds = manifold.MDS(2, max_iter=100, n_init=1, random_state=42)
trans_data = mds.fit_transform(sphere_data).T
t1 = time()
print("MDS: %.2g sec" % (t1 - t0))

ax = fig.add_subplot(258)
plt.scatter(trans_data[0], trans_data[1], c=colors, cmap=plt.cm.rainbow)
plt.title("MDS (%.2g sec)" % (t1 - t0))
ax.xaxis.set_major_formatter(NullFormatter())
ax.yaxis.set_major_formatter(NullFormatter())
plt.axis("tight")

# Perform Spectral Embedding.
t0 = time()
se = manifold.SpectralEmbedding(
    n_components=2, n_neighbors=n_neighbors, random_state=42
)
trans_data = se.fit_transform(sphere_data).T
t1 = time()
print("Spectral Embedding: %.2g sec" % (t1 - t0))

ax = fig.add_subplot(259)
plt.scatter(trans_data[0], trans_data[1], c=colors, cmap=plt.cm.rainbow)
plt.title("Spectral Embedding (%.2g sec)" % (t1 - t0))
ax.xaxis.set_major_formatter(NullFormatter())
ax.yaxis.set_major_formatter(NullFormatter())
plt.axis("tight")

# Perform t-distributed stochastic neighbor embedding.
t0 = time()
tsne = manifold.TSNE(n_components=2, random_state=0)
trans_data = tsne.fit_transform(sphere_data).T
t1 = time()
print("t-SNE: %.2g sec" % (t1 - t0))

ax = fig.add_subplot(2, 5, 10)
plt.scatter(trans_data[0], trans_data[1], c=colors, cmap=plt.cm.rainbow)
plt.title("t-SNE (%.2g sec)" % (t1 - t0))
ax.xaxis.set_major_formatter(NullFormatter())
ax.yaxis.set_major_formatter(NullFormatter())
plt.axis("tight")

plt.show()

脚本总运行时间：（0 分钟 7.884 秒）