理解 t-SNE （Python）

t-SNE（t-distribution Stochastic Neighbor Embedding）是目前最为流行的高维数据的降维算法。

t-SNE 成立的前提基于这样的一个假设：我们现实世界观察到的数据集，都在本质上有一种低维的特性（low intrinsic dimensionality），尽管它们嵌入在高维空间中，甚至可以说，高维数据经过降维后，在低维状态下，更能显现其本质特性，这其实也是流形学习（Manifold Learning）的基本思想。

原始论文请见，论文链接（pdf）。

1. sklearn 仿真

• import 必要的库；

  import numpy as np
  from numpy import linalg
  from numpy.linalg import norm
  from scipy.spatial.distance import squareform, pdist
  
  # We import sklearn.
  
  import sklearn
  from sklearn.manifold import TSNE
  from sklearn.datasets import load_digits
  from sklearn.preprocessing import scale
  
  # We'll hack a bit with the t-SNE code in sklearn 0.15.2.
  
  from sklearn.metrics.pairwise import pairwise_distances
  from sklearn.manifold.t_sne import (_joint_probabilities,
                                      _kl_divergence)
  from sklearn.utils.extmath import _ravel
  
  # Random state.
  
  RS = 20150101
  
  # We'll use matplotlib for graphics.
  
  import matplotlib.pyplot as plt
  import matplotlib.patheffects as PathEffects
  import matplotlib
  %matplotlib inline
  
  # We import seaborn to make nice plots.
  
  import seaborn as sns
  sns.set_style('darkgrid')
  sns.set_palette('muted')
  sns.set_context("notebook", font_scale=1.5,
                  rc={"lines.linewidth": 2.5})
  
  # We'll generate an animation with matplotlib and moviepy.
  
  from moviepy.video.io.bindings import mplfig_to_npimage
  import moviepy.editor as mpy

• 加载数据集

  digits = load_digits()
          # digits.data.shape ⇒ (1797L, 64L)

• 调用 sklearn 工具箱中的 t-SNE 类

  X = np.vstack([digits.data[digits.target==i]
                 for i in range(10)])
  y = np.hstack([digits.target[digits.target==i]
                 for i in range(10)])
  digits_proj = TSNE(random_state=RS).fit_transform(X)
          # digits_proj：(1797L, 2L)，ndarray 类型

• 可视化

  def scatter(x, colors):
      # We choose a color palette with seaborn.
      palette = np.array(sns.color_palette("hls", 10))
  
      # We create a scatter plot.
      f = plt.figure(figsize=(8, 8))
      ax = plt.subplot(aspect='equal')
      sc = ax.scatter(x[:,0], x[:,1], lw=0, s=40,
                      c=palette[colors.astype(np.int)])
      plt.xlim(-25, 25)
      plt.ylim(-25, 25)
      ax.axis('off')
      ax.axis('tight')
  
      # We add the labels for each digit.
      txts = []
      for i in range(10):
          # Position of each label.
          xtext, ytext = np.median(x[colors == i, :], axis=0)
          txt = ax.text(xtext, ytext, str(i), fontsize=24)
          txt.set_path_effects([
              PathEffects.Stroke(linewidth=5, foreground="w"),
              PathEffects.Normal()])
          txts.append(txt)
  
      return f, ax, sc, txts
  scatter(digits_proj, y)
  plt.savefig('images/digits_tsne-generated.png', dpi=120)

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An illustrated introduction to the t-SNE algorithm