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U, s, Vt = LA.svd(face_data, full_matrices=False)
S = np.diag(s)
#U: 500 *500, S: 500 *500, Vt:500 *2914

dim = 100 
US = U[:, 0:dim].dot(S[0:dim, 0:dim])

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truncated = faces.data[:500] # 500 *2914 Matrix
average_face = truncted.mean(axis=0)
face_data = truncted - average_face

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from sklearn.datasets import fetch_lfw_people

faces = fetch_lfw_people(min_faces_per_person=60)
face_data_raw = faces.data
average_face = face_data_raw.mean(axis=0)
face_data = face_data_raw - average_face

pca = PCA(100).fit(face_data)
PC = pca.transform(face_data)

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pca = PCA(2)
PC = pca.fit_transform(digits.data)
inversed = pca.inverse_transform(PC)
_, axes = plt.subplots(4, 10, figsize=(10, 4), 
                         subplot_kw={'xticks':[], 'yticks':[]},
                         gridspec_kw=dict(hspace=0.1, wspace=0.1))
for i, ax in enumerate(axes.flat):
    ax.imshow(inversed[i].reshape(8, 8),
              cmap='binary', interpolation='nearest', clim=(0, 16))

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_, axes = plt.subplots(2, 8, figsize=(8, 2), 
                       subplot_kw={'xticks':[], 'yticks':[]},
                       gridspec_kw=dict(hspace=0.1, wspace=0.1))
for i, ax in enumerate(axes.flat):
    ax.imshow(digits.data[i].reshape(8, 8),
              cmap='binary', interpolation='nearest', clim=(0, 16))

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import seaborn as sns
sns.set()
sns.despine(left=True)

fig, axes = plt.subplots(2, 2, figsize=(15, 10), gridspec_kw=dict(hspace=.4, wspace=.3))
title_settings={'fontsize':16}
subtitles=['Inversed samples with {} components']*3

# Plot Heatmap 1
ax = sns.heatmap(digits.data, cbar=False, ax=axes[0, 0])

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pca = PCA().fit(digits.data) # Blank inside the closed bracket
plt.plot(np.cumsum(pca.explained_variance_ratio_),'o-', c='#663399', alpha=.5)
plt.xlabel('Number of components')
plt.ylabel('Cumulative explained variance')

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plt.plot(np.cumsum(pca.explained_variance_ratio_),'o-', c='#663399', alpha=.5)
plt.xlabel('Number of components')
plt.ylabel('Cumulative explained variance')

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from sklearn.datasets import load_digits
digits = load_digits()
pca = PCA(n_components=10).fit(digits.data)

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ax = plt.gca(); ax.set_xlabel('X'); ax.set_ylabel('Y')
plt.scatter(X[:, 0], X[:, 1], alpha=0.3, color="#191970")
plt.scatter(pca.mean_[0], pca.mean_[1], c='red', s=50)
plt.axis('equal')

for length, vector in zip(pca.explained_variance_, pca.components_):
    dir_ = vector * 3  * np.sqrt(length)
    start = pca.mean_; end = start + dir_
    arrowprops = dict(arrowstyle='->',linewidth=2, 
                      shrinkA=0, shrinkB=0, color='red', alpha=0.5)