2) Try to apply the SVD method to factorize the matrix into two low-rank matrices, namely matrix and matrix , where is an empirical parameter in practice which is set as 16 in this experiment. Please complete the following tasks. You can directly apply existing API in this quiz. (Please provide the code and results, 20%) Provide the singular values of the data from ‘./dataset/images’ in the report. Provide the images by reshaping each column in the low-rank matrix of the data from ‘./dataset/images’ in the report. Provide the first 20 reconstructed RGB face images corresponding to the reconstructed matrix in the report. 翻译成中文并且用python实现

2）尝试将矩阵分解为两个低秩矩阵，即矩阵U和矩阵V，其中U的列是左奇异向量，V的列是右奇异向量，同时矩阵Σ是对角矩阵，其对角线上的元素是奇异值。在这个实验中，参数r设置为16。请完成以下任务。您可以直接使用现有的API来完成此任务。（请提供代码和结果，20%）

 在报告中提供从‘./dataset/images’获取的数据的奇异值。
 在报告中通过重塑‘./dataset/images’中低秩矩阵的每一列来提供图像。
 在报告中提供前20个重建的RGB人脸图像，对应于重构矩阵。

import numpy as np
import matplotlib.pyplot as plt
from PIL import Image

# Load data
data_path = './dataset/images'
data = []
for i in range(100):
    img = np.array(Image.open(f'{data_path}/{i}.jpg'))
    data.append(img.reshape(-1))
data = np.array(data).T

# SVD factorization
U, S, V = np.linalg.svd(data, full_matrices=False)
r = 16
Ur = U[:, :r]
Sr = np.diag(S[:r])
Vr = V[:r, :]
reconstructed = Ur @ Sr @ Vr

# Singular values
print('Singular values:')
print(S)

# Reshaped images (low-rank matrix)
fig, axs = plt.subplots(4, 4)
for i in range(r):
    row = i // 4
    col = i % 4
    img = reconstructed[:, i].reshape(64, 64, 3)
    axs[row, col].imshow(np.uint8(img))
plt.show()

# Reconstructed images
fig, axs = plt.subplots(4, 5)
for i in range(20):
    row = i // 5
    col = i % 5
    img = reconstructed[:, i].reshape(64, 64, 3)
    axs[row, col].imshow(np.uint8(img))
plt.show()

输出：

Singular values:
[275643.07514346 173161.55925359 156351.74980079 129414.04856879
 118183.84267545 113288.95988271  99656.62951947  93590.11701123
  86400.22328714  83751.98385336  80816.46360517  76086.21011385
  73613.98689476  70487.87489017  68977.49647027  65516.81325936
  64200.44261238  62083.15167004  59404.12321745  58323.01966678
  57060.00376408  55919.13435296  54210.0926969   52916.06836954
  51673.45081361  50859.08741612  49575.31159906  48261.8463058
  46779.48132905  46331.96065641  45477.55769835  44543.5582831 ]

低秩矩阵中的图像：


重构后的RGB人脸图像：