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Delete Book4_Ch16_Python_Codes directory
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Iris Series: Visualize Math -- From Arithmetic Basics to Machine Learning
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###############
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# Authored by Weisheng Jiang
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# Book 4 | From Basic Arithmetic to Machine Learning
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# Published and copyrighted by Tsinghua University Press
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# Beijing, China, 2022
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###############
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# Bk4_Ch16_01.py
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#%%
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# Bk4_Ch16_01_A
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import numpy as np
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from matplotlib import pyplot as plt
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import seaborn as sns
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PRECISION = 3
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def svd(X,full_matrices):
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U, s, Vt = np.linalg.svd(X,full_matrices = full_matrices)
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# Put the vector singular values into a padded matrix
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if full_matrices:
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S = np.zeros(X.shape)
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np.fill_diagonal(S, s)
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else:
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S = np.diag(s)
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# Rounding for display
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return np.round(U, PRECISION), np.round(S, PRECISION), np.round(Vt.T, PRECISION)
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# Repeatability
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np.random.seed(1)
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# Generate random matrix
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X = np.random.randn(6, 4)
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# manipulate X and reduce rank to 3
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# X[:,3] = X[:,0] + X[:,1]
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all_max = 2; all_min = -2
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#%% full
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U, S, V = svd(X, full_matrices = True)
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fig, axs = plt.subplots(1, 4, figsize=(12, 3))
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plt.sca(axs[0])
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ax = sns.heatmap(X,cmap='RdBu_r',vmax = all_max,vmin = all_min,
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cbar_kws={"orientation": "horizontal"})
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ax.set_aspect("equal")
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plt.title('$X$')
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plt.sca(axs[1])
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ax = sns.heatmap(U,cmap='RdBu_r',vmax = all_max,vmin = all_min,
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cbar_kws={"orientation": "horizontal"})
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ax.set_aspect("equal")
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plt.title('$U$')
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plt.sca(axs[2])
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ax = sns.heatmap(S,cmap='RdBu_r',vmax = all_max,vmin = all_min,
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cbar_kws={"orientation": "horizontal"})
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ax.set_aspect("equal")
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plt.title('$S$')
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plt.sca(axs[3])
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ax = sns.heatmap(V.T,cmap='RdBu_r',vmax = all_max,vmin = all_min,
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cbar_kws={"orientation": "horizontal"})
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ax.set_aspect("equal")
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plt.title('$V^T$')
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#%%
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# Bk4_Ch16_01_B
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#%% Economy-size, thin
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U, S, V = svd(X, full_matrices = False)
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fig, axs = plt.subplots(1, 4, figsize=(12, 3))
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plt.sca(axs[0])
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ax = sns.heatmap(X,cmap='RdBu_r',vmax = all_max,vmin = all_min,
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cbar_kws={"orientation": "horizontal"})
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ax.set_aspect("equal")
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plt.title('$X$')
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plt.sca(axs[1])
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ax = sns.heatmap(U,cmap='RdBu_r',vmax = all_max,vmin = all_min,
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cbar_kws={"orientation": "horizontal"})
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ax.set_aspect("equal")
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plt.title('$U$')
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plt.sca(axs[2])
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ax = sns.heatmap(S,cmap='RdBu_r',vmax = all_max,vmin = all_min,
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cbar_kws={"orientation": "horizontal"})
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ax.set_aspect("equal")
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plt.title('$S$')
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plt.sca(axs[3])
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ax = sns.heatmap(V.T,cmap='RdBu_r',vmax = all_max,vmin = all_min,
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cbar_kws={"orientation": "horizontal"})
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ax.set_aspect("equal")
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plt.title('$V^T$')
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#%%
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# Bk4_Ch16_01_C
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#%% Compact
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import copy
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X_rank_3 = copy.deepcopy(X);
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# manipulate X and reduce rank to 3
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X_rank_3[:,3] = X[:,0] + X[:,1]
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U_rank_3, S_rank_3, V_rank_3 = svd(X_rank_3, full_matrices = False)
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fig, axs = plt.subplots(1, 4, figsize=(12, 3))
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plt.sca(axs[0])
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ax = sns.heatmap(X_rank_3,cmap='RdBu_r',vmax = all_max,vmin = all_min,
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cbar_kws={"orientation": "horizontal"})
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ax.set_aspect("equal")
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plt.title('$X$')
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plt.sca(axs[1])
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ax = sns.heatmap(U_rank_3,cmap='RdBu_r',vmax = all_max,vmin = all_min,
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cbar_kws={"orientation": "horizontal"})
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ax.set_aspect("equal")
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plt.title('$U$')
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plt.sca(axs[2])
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ax = sns.heatmap(S_rank_3,cmap='RdBu_r',vmax = all_max,vmin = all_min,
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cbar_kws={"orientation": "horizontal"})
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ax.set_aspect("equal")
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plt.title('$S$')
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plt.sca(axs[3])
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ax = sns.heatmap(V_rank_3.T,cmap='RdBu_r',vmax = all_max,vmin = all_min,
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cbar_kws={"orientation": "horizontal"})
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ax.set_aspect("equal")
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plt.title('$V^T$')
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#%%
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# Bk4_Ch16_01_D
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#%% Truncated
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num_p = 2;
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U_truc = U[:,0:num_p]
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S_truc = S[0:num_p,0:num_p]
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V_truc = V[:,0:num_p]
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X_hat = U_truc@S_truc@(V_truc.T)
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# reproduce
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fig, axs = plt.subplots(1, 4, figsize=(12, 3))
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plt.sca(axs[0])
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ax = sns.heatmap(X_hat,cmap='RdBu_r',vmax = all_max,vmin = all_min,
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cbar_kws={"orientation": "horizontal"})
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ax.set_aspect("equal")
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plt.title('$\hat{X}$')
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plt.sca(axs[1])
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ax = sns.heatmap(U_truc,cmap='RdBu_r',vmax = all_max,vmin = all_min,
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cbar_kws={"orientation": "horizontal"})
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ax.set_aspect("equal")
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plt.title('$U$')
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plt.sca(axs[2])
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ax = sns.heatmap(S_truc,cmap='RdBu_r',vmax = all_max,vmin = all_min,
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cbar_kws={"orientation": "horizontal"})
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ax.set_aspect("equal")
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plt.title('$S$')
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plt.sca(axs[3])
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ax = sns.heatmap(V_truc.T,cmap='RdBu_r',vmax = all_max,vmin = all_min,
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cbar_kws={"orientation": "horizontal"})
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ax.set_aspect("equal")
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plt.title('$V^T$')
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# Error
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fig, axs = plt.subplots(1, 3, figsize=(12, 3))
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plt.sca(axs[0])
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ax = sns.heatmap(X,cmap='RdBu_r',vmax = all_max,vmin = all_min,
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cbar_kws={"orientation": "horizontal"})
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ax.set_aspect("equal")
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plt.title('$X$')
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plt.sca(axs[1])
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ax = sns.heatmap(X_hat,cmap='RdBu_r',vmax = all_max,vmin = all_min,
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cbar_kws={"orientation": "horizontal"})
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ax.set_aspect("equal")
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plt.title('$\hat{X}$')
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plt.sca(axs[2])
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ax = sns.heatmap(X - X_hat,cmap='RdBu_r',vmax = all_max,vmin = all_min,
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cbar_kws={"orientation": "horizontal"})
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ax.set_aspect("equal")
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plt.title('$E = X - \hat{X}$')
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#%%
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# Bk4_Ch16_01_E
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#%% Tensor products
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u1_outer_v1 = np.outer(U[:,0][:, None], V[:,0][:, None]);
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u2_outer_v2 = np.outer(U[:,1][:, None], V[:,1][:, None]);
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u3_outer_v3 = np.outer(U[:,2][:, None], V[:,2][:, None]);
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u4_outer_v4 = np.outer(U[:,3][:, None], V[:,3][:, None]);
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# visualize tensor products
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fig, axs = plt.subplots(1, 4, figsize=(12, 3))
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plt.sca(axs[0])
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ax = sns.heatmap(u1_outer_v1,cmap='RdBu_r',vmax = all_max,vmin = all_min,
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cbar_kws={"orientation": "horizontal"})
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ax.set_aspect("equal")
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plt.title('$u1v1^T$')
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plt.sca(axs[1])
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ax = sns.heatmap(u2_outer_v2,cmap='RdBu_r',vmax = all_max,vmin = all_min,
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cbar_kws={"orientation": "horizontal"})
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ax.set_aspect("equal")
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plt.title('$u2v2^T$')
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plt.sca(axs[2])
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ax = sns.heatmap(u3_outer_v3,cmap='RdBu_r',vmax = all_max,vmin = all_min,
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cbar_kws={"orientation": "horizontal"})
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ax.set_aspect("equal")
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plt.title('$u3v3^T$')
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plt.sca(axs[3])
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ax = sns.heatmap(u4_outer_v4,cmap='RdBu_r',vmax = all_max,vmin = all_min,
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cbar_kws={"orientation": "horizontal"})
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ax.set_aspect("equal")
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plt.title('$u4v4^T$')
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X_1 = S[0,0]*u1_outer_v1
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# X_1 = S[0,0]*U[:,0][:, None]@V[:,0][None, :];
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X_2 = S[1,1]*u2_outer_v2
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# X_2 = S[1,1]*U[:,1][:, None]@V[:,1][None, :];
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X_3 = S[2,2]*u3_outer_v3
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# X_3 = S[2,2]*U[:,2][:, None]@V[:,2][None, :];
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X_4 = S[3,3]*u4_outer_v4
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# X_4 = S[3,3]*U[:,3][:, None]@V[:,3][None, :];
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# visualize components
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fig, axs = plt.subplots(1, 4, figsize=(12, 3))
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plt.sca(axs[0])
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ax = sns.heatmap(X_1,cmap='RdBu_r',vmax = all_max,vmin = all_min,
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cbar_kws={"orientation": "horizontal"})
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ax.set_aspect("equal")
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plt.title('$\hat{X}_1$')
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plt.sca(axs[1])
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ax = sns.heatmap(X_2,cmap='RdBu_r',vmax = all_max,vmin = all_min,
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cbar_kws={"orientation": "horizontal"})
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ax.set_aspect("equal")
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plt.title('$\hat{X}_2$')
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plt.sca(axs[2])
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ax = sns.heatmap(X_3,cmap='RdBu_r',vmax = all_max,vmin = all_min,
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cbar_kws={"orientation": "horizontal"})
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ax.set_aspect("equal")
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plt.title('$\hat{X}_3$')
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plt.sca(axs[3])
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ax = sns.heatmap(X_4,cmap='RdBu_r',vmax = all_max,vmin = all_min,
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cbar_kws={"orientation": "horizontal"})
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ax.set_aspect("equal")
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plt.title('$\hat{X}_4$')
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#%%
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# Bk4_Ch16_01_F
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#%% Reproduction and error
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fig, axs = plt.subplots(1, 3, figsize=(12, 3))
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plt.sca(axs[0])
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ax = sns.heatmap(X,cmap='RdBu_r',vmax = all_max,vmin = all_min,
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cbar_kws={"orientation": "horizontal"})
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ax.set_aspect("equal")
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plt.title('$X$')
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plt.sca(axs[1])
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ax = sns.heatmap(X_1 + X_2,cmap='RdBu_r',vmax = all_max,vmin = all_min,
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cbar_kws={"orientation": "horizontal"})
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ax.set_aspect("equal")
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plt.title('$\hat{X}_1 + \hat{X}_2$')
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plt.sca(axs[2])
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ax = sns.heatmap(X - (X_1 + X_2),cmap='RdBu_r',vmax = all_max,vmin = all_min,
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cbar_kws={"orientation": "horizontal"})
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ax.set_aspect("equal")
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plt.title('$X - (\hat{X}_1 + \hat{X}_2)$')
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fig, axs = plt.subplots(1, 3, figsize=(12, 3))
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plt.sca(axs[0])
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ax = sns.heatmap(X,cmap='RdBu_r',vmax = all_max,vmin = all_min,
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cbar_kws={"orientation": "horizontal"})
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ax.set_aspect("equal")
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plt.title('$X$')
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plt.sca(axs[1])
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ax = sns.heatmap(X_1 + X_2 + X_3,cmap='RdBu_r',vmax = all_max,vmin = all_min,
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cbar_kws={"orientation": "horizontal"})
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ax.set_aspect("equal")
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plt.title('$\hat{X}_1 + \hat{X}_2 + \hat{X}_3$')
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plt.sca(axs[2])
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ax = sns.heatmap(X - (X_1 + X_2 + X_3),cmap='RdBu_r',vmax = all_max,vmin = all_min,
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cbar_kws={"orientation": "horizontal"})
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ax.set_aspect("equal")
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plt.title('$X - (\hat{X}_1 + \hat{X}_2 + \hat{X}_3)$')
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@@ -1,87 +0,0 @@
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###############
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# Authored by Weisheng Jiang
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# Book 4 | From Basic Arithmetic to Machine Learning
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# Published and copyrighted by Tsinghua University Press
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# Beijing, China, 2022
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###############
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import streamlit as st
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import plotly.express as px
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import numpy as np
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import pandas as pd
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from sklearn.datasets import load_iris
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#%%
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# A copy from Seaborn
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iris = load_iris()
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X = iris.data
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y = iris.target
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feature_names = ['Sepal length, x1','Sepal width, x2',
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'Petal length, x3','Petal width, x4']
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# Convert X array to dataframe
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X_df = pd.DataFrame(X, columns=feature_names)
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with st.sidebar:
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st.latex('X = USV^T')
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st.latex('X = \sum_{j=1}^{D} s_j u_j v_j^T')
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st.latex('X \simeq \sum_{j=1}^{p} s_j u_j v_j^T')
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#%% Original data, X
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X = X_df.to_numpy();
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U, S, V_T = np.linalg.svd(X, full_matrices=False)
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S = np.diag(S)
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V = V_T.T
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with st.sidebar:
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p = st.slider('Choose p, number of component to approximate X:',
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1,4,step = 1)
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#%% Approximate X
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X_apprx = U[:,0:p]@S[0:p,0:p]@V[:,0:p].T
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X_apprx_df = pd.DataFrame(X_apprx, columns = feature_names)
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Error_df = X_df - X_apprx_df
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#%%
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col1, col2, col3 = st.columns(3)
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with col1:
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st.latex('X')
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fig_1 = px.imshow(X_df,
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color_continuous_scale='RdYlBu_r',
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range_color = [0,8])
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fig_1.layout.height = 500
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fig_1.layout.width = 300
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fig_1.update_layout(coloraxis_showscale=False)
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st.plotly_chart(fig_1)
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with col2:
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st.latex('\hat{X}')
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fig_2 = px.imshow(X_apprx_df,
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color_continuous_scale='RdYlBu_r',
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range_color = [0,8])
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fig_2.layout.height = 500
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fig_2.layout.width = 300
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fig_2.update_layout(coloraxis_showscale=False)
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st.plotly_chart(fig_2)
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with col3:
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st.latex('X - \hat{X}')
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fig_3 = px.imshow(Error_df,
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color_continuous_scale='RdYlBu_r',
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range_color = [0,8])
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fig_3.layout.height = 500
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fig_3.layout.width = 300
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fig_3.update_layout(coloraxis_showscale=False)
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st.plotly_chart(fig_3)
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Reference in New Issue
Block a user