SVD奇异值分解与PCA主成分分析

SVD动画图解–Wiki

Eigen Svd 和 np.linalg.svd都可以用于SVD计算，但两者却存在细微的差别。

python

import numpy as np
data=np.array([[0.99337785, 0.08483806, 0.07747866, -92.91055059],[-0.07889607, 0.99392169, -0.07677948, -42.2437898],[-0.08352154, 0.07015827, 0.99403318, 396.22910711],[0, 0, 0, 1]])
U, S, Vt = np.linalg.svd(data)
k = 4
U_reduced = U[:, :k]
S_reduced = np.diag(S[:k])
Vt_reduced = Vt[:k, :]# 重构原始数据
# reconstructed_data = np.dot(U_reduced, np.dot(S_reduced, Vt_reduced))
reconstructed_data = np.dot(U*S, Vt)
print(f"原始数据:\n{data}")
print(f"重构的数据:\n{reconstructed_data}")
print(f"U:\n{U_reduced}")
print(f"S:\n{S}")
print(f"Vt:\n{Vt_reduced}")np.allclose(data, reconstructed_data, rtol=1e-05, atol=1e-08) #true

原始数据:
[[ 0.99337785 0.08483806 0.07747866 -92.91055059]
[ -0.07889607 0.99392169 -0.07677948 -42.2437898 ]
[ -0.08352154 0.07015827 0.99403318 396.22910711]
[ 0. 0. 0. 1. ]]
重构的数据:
[[ 0.99337785 0.08483806 0.07747866 -92.91055059]
[ -0.07889607 0.99392169 -0.07677948 -42.2437898 ]
[ -0.08352154 0.07015827 0.99403318 396.22910711]
[ -0. 0. -0. 1. ]]
U:
[[-0.22707395 0.44440135 0.86657057 -0.00055497]
[-0.10324408 0.87381543 -0.47517069 -0.00025233]
[ 0.96838634 0.19736775 0.15253937 0.00236674]
[ 0.00244399 0. -0. -0.99999701]]
S:
[409.16550869 1. 0.99999999 0.002444 ]
Vt:
[[-0.00072906 -0.00013183 0.00232899 0.99999701]
[ 0.35603339 0.92005324 0.16353062 0. ]
[ 0.88558079 -0.38806239 0.25525328 -0. ]
[-0.2983058 -0.05394073 0.9529418 -0.00244399]]
True

c++

// 定义给定的变换矩阵
Eigen::Matrix4f transformation_matrix1;
transformation_matrix1 << 0.99337785, 0.08483806, 0.07747866, -92.91055059,-0.07889607, 0.99392169, -0.07677948, -42.2437898,-0.08352154, 0.07015827, 0.99403318, 396.22910711,0, 0, 0, 1;// 从变换矩阵中解析欧拉角
Eigen::Vector3f euler_angles1 = transformation_matrix1.block<3, 3>(0, 0).eulerAngles(0, 2, 1);// 输出解析出的欧拉角
std::cout << "解析出的欧拉角 (ZYX 顺序):"<<euler_angles1 << std::endl;Eigen::JacobiSVD<Eigen::MatrixXf> svd;
//svd.setThreshold(1e-10);
svd.compute(transformation_matrix1, Eigen::ComputeFullU | Eigen::ComputeFullV);
Eigen::VectorXf singularValues = svd.singularValues();
Eigen::MatrixXf singularValueMatrix = singularValues.asDiagonal();
Eigen::MatrixXf U = svd.matrixU();
Eigen::MatrixXf V = svd.matrixV();
//Displaying the results
std::cout << "原始数据:\n" << transformation_matrix1  << std::endl;
std::cout << "重构数据:\n" << U * singularValueMatrix * V.transpose() << std::endl;
std::cout << "U:\n" << U << "\n\n";
std::cout <<  "S:\n" << singularValueMatrix << "\n\n";
std::cout << "V:\n" << V << "\n\n";

原始数据:
0.993378 0.0848381 0.0774787 -92.9106
-0.0788961 0.993922 -0.0767795 -42.2438
-0.0835215 0.0701583 0.994033 396.229
0 0 0 1
重构数据:
0.9934 0.0848412 0.077413 -92.9106
-0.0788853 0.993924 -0.0768136 -42.2438
-0.0835193 0.070161 0.994014 396.229
0 9.21136e-09 2.32831e-10 1
U:
-0.227074 0.457935 0.859496 -0.000554795
-0.103244 -0.888896 0.446323 -0.000252237
0.968386 0.0126106 0.249125 0.00236679
0.00244399 8.88704e-10 2.06488e-07 -0.999997
S:
409.166 0 0 0
0 1 0 0
0 0 1 0
0 0 0 0.002444
V:
-0.000729068 0.52398 0.797807 -0.298242
-0.000131827 -0.843758 0.534011 -0.0538979
0.00232899 0.116264 0.279886 0.952964
0.999997 3.72529e-09 2.01166e-07 -0.00244399

结论

$$M=U\Sigma V^T$$

当分解的矩阵M相同时，不同库得出的U,$\Sigma$是相同的，V却不相同。numpy给出的直接就是V^T,而EIGEN得出的为V.

参考

1. https://eigen.tuxfamily.org/dox/group__SVD__Module.html
2. https://numpy.org/doc/stable/reference/generated/numpy.linalg.svd.html
3. https://stackoverflow.com/questions/75500005/linalg-svd-and-jacobisvdmatrixxf-svd-the-results-are-different
4. https://stackoverflow.com/questions/74322089/difference-between-eigen-svd-and-np-linag-svd
5. https://en.wikipedia.org/wiki/Singular_value_decomposition