数据科学【八】：SVD（一）

本文旨在给出SVD的使用方法。具体原理或SVD本身的代码实现请参考其他资料。
SVD主要应用于数据特征提取，数据压缩等。

数据准备

将mnist存入csv

使用fetch_openml可以获得常用数据集，包括mnist_784。

import matplotlib.pyplot as plt

from sklearn.datasets import fetch_openml

X, y = fetch_openml(name="mnist_784", version=1, return_X_y=True, as_frame=False)
  
import pandas as pd
import numpy as np


full_data = np.c_[y, X]
full_df = pd.DataFrame(full_data)
full_df.to_csv("mnist.csv", index=False)

获得特征值

获得某个“0”的特征值

SVD可以调用numpy的linalg.svd。

import matplotlib.pyplot as plt
full_df = pd.read_csv("mnist.csv", low_memory = False)
full_data = full_df.values
plt.figure()

for n in range(100):
    if full_data[n][0] == 0:
        print(n)
        data = full_data[n][1:].reshape(28, 28)
        u, s, v = np.linalg.svd(data)
        plt.plot(s)
        break

plt.show()

数据压缩

我们可以通过保留部分特征值来进行数据的压缩或模糊。

单图片压缩

示例：通过将一些特征值设置为0，将“0”的图片压缩。
矩阵乘法可以通过numpy.matmul实现。

def image_svd(n, data):
    u, s, v = np.linalg.svd(data)
    svd = np.zeros((u.shape[0], v.shape[1]))
    for i in range(n):
        svd[i, i] = s[i]
    img = np.matmul(u, svd)
    img = np.matmul(img, v)
    return img

plt.figure()
plt.subplot(1, 2, 1)
original_img = full_data[1][1:].reshape(28, 28)
plt.imshow(original_img, cmap="gray")
compress_img = image_svd(10, original_img)
plt.subplot(1, 2, 2)
plt.imshow(compress_img, cmap="gray")
plt.show()

全数据集压缩

示例：将整个数据集采用以上方法压缩并存入csv

X_data = full_data[:, range(1, 785)]
app_X = np.zeros(X_data.shape)
for i in range(X_data.shape[1]):
    original = X_data[i].reshape(28, 28)
    svd_img = image_svd(10, original)
    app_X[i] = svd_img.reshape(1,784)
app_df = pd.DataFrame(app_X)
app_df.to_csv("app_mnist.csv", index=False)

一些现象

聚类离散性

首先，我们采用与上一篇同样的方法，对数据集进行十聚类并绘制聚类中心：

from sklearn.cluster import KMeans
from sklearn.decomposition import PCA
import pandas as pd
import matplotlib.pyplot as plt

app_df = pd.read_csv("app_mnist.csv", low_memory=False)
kmeans_app = KMeans(n_clusters=10)
kmeans_app.fit(app_df.values)
app_centers = kmeans_app.cluster_centers_
centers_2d = PCA(2).fit_transform(app_centers)
plt.scatter(centers_2d[:, 0], centers_2d[:, 1])
plt.show()

我们在压缩后的数据集上再来一遍：

org_df = pd.read_csv("mnist.csv", low_memory=False)
kmeans_org = KMeans(n_clusters=10)
kmeans_org.fit(org_df.values)
org_centers = kmeans_org.cluster_centers_
centers_2d = PCA(2).fit_transform(org_centers)
plt.scatter(centers_2d[:, 0], centers_2d[:, 1])
plt.show()

聚类准确性

分别计算原数据集与压缩数据集相对于ground truth的disagreement distance：

def disagreement_dist(P_labels, C_labels):
    answer = 0
    for i in range(len(P_labels)-1):
        for j in range(i+1, len(P_labels)):
            if (P_labels[i] == P_labels[j]) != (C_labels[i]==C_labels[j]):
                answer += 1
    return answer

import numpy as np
org_plabels = kmeans_org.labels_
app_plabels = kmeans_app.labels_
clabels = pd.read_csv("mnist.csv", low_memory=False).values[:, [0]]
clabels.reshape(1, len(clabels))
clabels.astype(np.int8)
print("Difference on original dataset:")
print(disagreement_dist(org_plabels, clabels))
print("Difference on approximated dataset:")
print(disagreement_dist(app_plabels, clabels))

Difference on original dataset:
288675700
Difference on approximated dataset:
2161020505