Johnson-Lindenstrauss绑定为带有随机投影的嵌入简介

代码实现[Python]

# -*- coding: utf-8 -*- 
print(__doc__)

import sys
from time import time
import numpy as np
import matplotlib
import matplotlib.pyplot as plt
from distutils.version import LooseVersion
from sklearn.random_projection import johnson_lindenstrauss_min_dim
from sklearn.random_projection import SparseRandomProjection
from sklearn.datasets import fetch_20newsgroups_vectorized
from sklearn.datasets import load_digits
from sklearn.metrics.pairwise import euclidean_distances

# `normed` is being deprecated in favor of `density` in histograms
if LooseVersion(matplotlib.__version__) >= '2.1':
    density_param = {'density': True}
else:
    density_param = {'normed': True}

# Part 1: 绘制n_components_min和n_samples之间的理论依赖性

# 容许失真的范围
eps_range = np.linspace(0.1, 0.99, 5)
colors = plt.cm.Blues(np.linspace(0.3, 1.0, len(eps_range)))

# range of number of samples (observation) to embed
n_samples_range = np.logspace(1, 9, 9)

plt.figure()
for eps, color in zip(eps_range, colors):
    min_n_components = johnson_lindenstrauss_min_dim(n_samples_range, eps=eps)
    plt.loglog(n_samples_range, min_n_components, color=color)

plt.legend(["eps = %0.1f" % eps for eps in eps_range], loc="lower right")
plt.xlabel("Number of observations to eps-embed")
plt.ylabel("Minimum number of dimensions")
plt.title("Johnson-Lindenstrauss bounds:\nn_samples vs n_components")

# 容许失真的范围
eps_range = np.linspace(0.01, 0.99, 100)

# range of number of samples (observation) to embed
n_samples_range = np.logspace(2, 6, 5)
colors = plt.cm.Blues(np.linspace(0.3, 1.0, len(n_samples_range)))

plt.figure()
for n_samples, color in zip(n_samples_range, colors):
    min_n_components = johnson_lindenstrauss_min_dim(n_samples, eps=eps_range)
    plt.semilogy(eps_range, min_n_components, color=color)

plt.legend(["n_samples = %d" % n for n in n_samples_range], loc="upper right")
plt.xlabel("Distortion eps")
plt.ylabel("Minimum number of dimensions")
plt.title("Johnson-Lindenstrauss bounds:\nn_components vs eps")


# Part 2: 对维数很低且密度高的某些数字图像或对维数高且稀疏的20个新闻组数据集执行稀疏随机投影
if '--twenty-newsgroups' in sys.argv:
    # Need an internet connection hence not enabled by default
    data = fetch_20newsgroups_vectorized().data[:500]
else:
    data = load_digits().data[:500]

n_samples, n_features = data.shape
print("Embedding %d samples with dim %d using various random projections"
      % (n_samples, n_features))

n_components_range = np.array([300, 1000, 10000])
dists = euclidean_distances(data, squared=True).ravel()

# 仅选择不相同的样本对
nonzero = dists != 0
dists = dists[nonzero]

for n_components in n_components_range:
    t0 = time()
    rp = SparseRandomProjection(n_components=n_components)
    projected_data = rp.fit_transform(data)
    print("Projected %d samples from %d to %d in %0.3fs"
          % (n_samples, n_features, n_components, time() - t0))
    if hasattr(rp, 'components_'):
        n_bytes = rp.components_.data.nbytes
        n_bytes += rp.components_.indices.nbytes
        print("Random matrix with size: %0.3fMB" % (n_bytes / 1e6))

    projected_dists = euclidean_distances(
        projected_data, squared=True).ravel()[nonzero]

    plt.figure()
    plt.hexbin(dists, projected_dists, gridsize=100, cmap=plt.cm.PuBu)
    plt.xlabel("Pairwise squared distances in original space")
    plt.ylabel("Pairwise squared distances in projected space")
    plt.title("Pairwise distances distribution for n_components=%d" %
              n_components)
    cb = plt.colorbar()
    cb.set_label('Sample pairs counts')

    rates = projected_dists / dists
    print("Mean distances rate: %0.2f (%0.2f)"
          % (np.mean(rates), np.std(rates)))

    plt.figure()
    plt.hist(rates, bins=50, range=(0., 2.), edgecolor='k', **density_param)
    plt.xlabel("Squared distances rate: projected / original")
    plt.ylabel("Distribution of samples pairs")
    plt.title("Histogram of pairwise distance rates for n_components=%d" %
              n_components)

    # TODO: compute the expected value of eps and add them to the previous plot
    # as vertical lines / region

plt.show()

代码执行

代码运行时间大约:0分1.837秒。
运行代码输出的文本内容如下:

Embedding 500 samples with dim 64 using various random projections
Projected 500 samples from 64 to 300 in 0.016s
Random matrix with size: 0.028MB
Mean distances rate: 0.97 (0.08)
Projected 500 samples from 64 to 1000 in 0.048s
Random matrix with size: 0.096MB
Mean distances rate: 0.99 (0.05)
Projected 500 samples from 64 to 10000 in 0.594s
Random matrix with size: 0.964MB
Mean distances rate: 1.01 (0.01)

运行代码输出的图片内容如下:

源码下载

• Python版源码文件: plot_johnson_lindenstrauss_bound.py
• Jupyter Notebook版源码文件: plot_johnson_lindenstrauss_bound.ipynb

参考资料

• The Johnson-Lindenstrauss bound for embedding with random projections