矩阵奇异值分解简介及C++/OpenCV/Eigen的三种实现

奇异值分解(singular value decomposition, SVD)：将矩阵分解为奇异向量(singular vector)和奇异值(singular value)。通过奇异值分解，我们会得到一些与特征分解相同类型的信息。然而，奇异值分解有更广泛的应用。每个实数矩阵都有一个奇异值分解，但不一定都有特征分解。例如，非方阵的矩阵没有特征分解，这是我们只能使用奇异值分解。

将矩阵A分解成三个矩阵的乘积：A=UDV^T。

假设A是一个m*n的矩阵，那么U是一个m*m的矩阵，D是一个m*n的矩阵，V是一个n*n矩阵。

这些矩阵中的每一个经定义后都拥有特殊的结构。矩阵U和V都被定义为正交矩阵，而矩阵D被定义为对角矩阵。注意，矩阵D不一定是方阵。对角矩阵D对角线上的元素被称为矩阵A的奇异值(singular value)。矩阵U的列向量被称为左奇异向量(left singular vector)，矩阵V的列向量被称为右奇异向量(right singular vector)。A的左奇异向量是AA^T的特征向量。A的右奇异向量是A^TA的特征向量。A的非零奇异值是A^TA特征值的平方根，同时也是AA^T特征值的平方根。

奇异值分解(singular value decomposition)是线性代数中一种重要的矩阵分解，在信号处理、统计学等领域有重要应用。奇异值分解能够用于任意m*n矩阵，而特征分解只能适用于特定类型的方阵，故奇异值分解的适用范围更广。

以上内容摘自： 《深度学习中文版》 和  维基百科

基于雅克比(Jacobi)方法对矩阵进行奇异值分解操作步骤(参考OpenCV中实现)：

(1)、对原始矩阵A(m*n)进行判断，若m<n(即行数小于列数)，则交换m、n值；若m≥n，则对A进行转置变换；

(2)、初始化临时变量D、U、Vt：D为奇异值，为n行1列；U为左奇异向量，为m行m列；Vt为转置后的右奇异向量，为n行n列；并初始化D、U、Vt值均为0；

(3)、初始化临时变量A′：A′为m行m列，并将A的值赋值给A′，A′中多余元素赋初值为0；

(4)、由A′、D、Vt开始进行基于Jacobi方法的奇异值分解；

(5)、设置临时变量W，长度为n，将A′中前n行中，每行元素的平方和赋值给W；

(6)、设置Vt为单位矩阵；

(7)、循环计算旋转矩阵，并更新A′、W、Vt对应位置的值；最大循环次数为std::max(m, 30)；

(8)、重置W值为A′中前n行，每行元素平方和的开方；

(9)、将W中元素按照从大到小排序，排序后的W即为D中主对角线元素值；

(10)、按照(9)中对W的排序规则对A′和Vt进行排序；

(11)、计算A′中值；

(12)、最终的A′和Vt即为所求的左、右奇异向量。

以下是分别采用C++(参考opencv sources/modules/core/src/lapack.cpp)和OpenCV实现的矩阵奇异值分解：

#include "funset.hpp"
#include <math.h>
#include <iostream>
#include <string>
#include <vector>
#include <opencv2/opencv.hpp>
#include "common.hpp"// ================================= 矩阵奇异值分解 =================================
template<typename _Tp>
static void JacobiSVD(std::vector<std::vector<_Tp>>& At,std::vector<std::vector<_Tp>>& _W, std::vector<std::vector<_Tp>>& Vt)
{double minval = FLT_MIN;_Tp eps = (_Tp)(FLT_EPSILON * 2);const int m = At[0].size();const int n = _W.size();const int n1 = m; // urowsstd::vector<double> W(n, 0.);for (int i = 0; i < n; i++) {double sd{0.};for (int k = 0; k < m; k++) {_Tp t = At[i][k];sd += (double)t*t;}W[i] = sd;for (int k = 0; k < n; k++)Vt[i][k] = 0;Vt[i][i] = 1;}int max_iter = std::max(m, 30);for (int iter = 0; iter < max_iter; iter++) {bool changed = false;_Tp c, s;for (int i = 0; i < n - 1; i++) {for (int j = i + 1; j < n; j++) {_Tp *Ai = At[i].data(), *Aj = At[j].data();double a = W[i], p = 0, b = W[j];for (int k = 0; k < m; k++)p += (double)Ai[k] * Aj[k];if (std::abs(p) <= eps * std::sqrt((double)a*b))continue;p *= 2;double beta = a - b, gamma = hypot_((double)p, beta);if (beta < 0) {double delta = (gamma - beta)*0.5;s = (_Tp)std::sqrt(delta / gamma);c = (_Tp)(p / (gamma*s * 2));} else {c = (_Tp)std::sqrt((gamma + beta) / (gamma * 2));s = (_Tp)(p / (gamma*c * 2));}a = b = 0;for (int k = 0; k < m; k++) {_Tp t0 = c*Ai[k] + s*Aj[k];_Tp t1 = -s*Ai[k] + c*Aj[k];Ai[k] = t0; Aj[k] = t1;a += (double)t0*t0; b += (double)t1*t1;}W[i] = a; W[j] = b;changed = true;_Tp *Vi = Vt[i].data(), *Vj = Vt[j].data();for (int k = 0; k < n; k++) {_Tp t0 = c*Vi[k] + s*Vj[k];_Tp t1 = -s*Vi[k] + c*Vj[k];Vi[k] = t0; Vj[k] = t1;}}}if (!changed)break;}for (int i = 0; i < n; i++) {double sd{ 0. };for (int k = 0; k < m; k++) {_Tp t = At[i][k];sd += (double)t*t;}W[i] = std::sqrt(sd);}for (int i = 0; i < n - 1; i++) {int j = i;for (int k = i + 1; k < n; k++) {if (W[j] < W[k])j = k;}if (i != j) {std::swap(W[i], W[j]);for (int k = 0; k < m; k++)std::swap(At[i][k], At[j][k]);for (int k = 0; k < n; k++)std::swap(Vt[i][k], Vt[j][k]);}}for (int i = 0; i < n; i++)_W[i][0] = (_Tp)W[i];srand(time(nullptr));for (int i = 0; i < n1; i++) {double sd = i < n ? W[i] : 0;for (int ii = 0; ii < 100 && sd <= minval; ii++) {// if we got a zero singular value, then in order to get the corresponding left singular vector// we generate a random vector, project it to the previously computed left singular vectors,// subtract the projection and normalize the difference.const _Tp val0 = (_Tp)(1. / m);for (int k = 0; k < m; k++) {unsigned int rng = rand() % 4294967295; // 2^32 - 1_Tp val = (rng & 256) != 0 ? val0 : -val0;At[i][k] = val;}for (int iter = 0; iter < 2; iter++) {for (int j = 0; j < i; j++) {sd = 0;for (int k = 0; k < m; k++)sd += At[i][k] * At[j][k];_Tp asum = 0;for (int k = 0; k < m; k++) {_Tp t = (_Tp)(At[i][k] - sd*At[j][k]);At[i][k] = t;asum += std::abs(t);}asum = asum > eps * 100 ? 1 / asum : 0;for (int k = 0; k < m; k++)At[i][k] *= asum;}}sd = 0;for (int k = 0; k < m; k++) {_Tp t = At[i][k];sd += (double)t*t;}sd = std::sqrt(sd);}_Tp s = (_Tp)(sd > minval ? 1 / sd : 0.);for (int k = 0; k < m; k++)At[i][k] *= s;}
}// matSrc为原始矩阵，支持非方阵，matD存放奇异值，matU存放左奇异向量，matVt存放转置的右奇异向量
template<typename _Tp>
int svd(const std::vector<std::vector<_Tp>>& matSrc,std::vector<std::vector<_Tp>>& matD, std::vector<std::vector<_Tp>>& matU, std::vector<std::vector<_Tp>>& matVt)
{int m = matSrc.size();int n = matSrc[0].size();for (const auto& sz : matSrc) {if (n != sz.size()) {fprintf(stderr, "matrix dimension dismatch\n");return -1;}}bool at = false;if (m < n) {std::swap(m, n);at = true;}matD.resize(n);for (int i = 0; i < n; ++i) {matD[i].resize(1, (_Tp)0);}matU.resize(m);for (int i = 0; i < m; ++i) {matU[i].resize(m, (_Tp)0);}matVt.resize(n);for (int i = 0; i < n; ++i) {matVt[i].resize(n, (_Tp)0);}std::vector<std::vector<_Tp>> tmp_u = matU, tmp_v = matVt;std::vector<std::vector<_Tp>> tmp_a, tmp_a_;if (!at)transpose(matSrc, tmp_a);elsetmp_a = matSrc;if (m == n) {tmp_a_ = tmp_a;} else {tmp_a_.resize(m);for (int i = 0; i < m; ++i) {tmp_a_[i].resize(m, (_Tp)0);}for (int i = 0; i < n; ++i) {tmp_a_[i].assign(tmp_a[i].begin(), tmp_a[i].end());}}JacobiSVD(tmp_a_, matD, tmp_v);if (!at) {transpose(tmp_a_, matU);matVt = tmp_v;} else {transpose(tmp_v, matU);matVt = tmp_a_;}return 0;
}int test_SVD()
{//std::vector<std::vector<float>> vec{ { 1.2f, 2.5f, 5.6f, -2.5f },//				{ -3.6f, 9.2f, 0.5f, 7.2f },//				{ 4.3f, 1.3f, 9.4f, -3.4f },//				{ 6.4f, 0.1f, -3.7f, 0.9f } };//const int rows{ 4 }, cols{ 4 };//std::vector<std::vector<float>> vec{ { 1.2f, 2.5f, 5.6f, -2.5f },//				{ -3.6f, 9.2f, 0.5f, 7.2f },//				{ 4.3f, 1.3f, 9.4f, -3.4f } };//const int rows{ 3 }, cols{ 4 };std::vector<std::vector<float>> vec{ { 0.68f, 0.597f },{ -0.211f, 0.823f },{ 0.566f, -0.605f } };const int rows{ 3 }, cols{ 2 };fprintf(stderr, "source matrix:\n");print_matrix(vec);fprintf(stderr, "\nc++ implement singular value decomposition:\n");std::vector<std::vector<float>> matD, matU, matVt;if (svd(vec, matD, matU, matVt) != 0) {fprintf(stderr, "C++ implement singular value decomposition fail\n");return -1;}fprintf(stderr, "singular values:\n");print_matrix(matD);fprintf(stderr, "left singular vectors:\n");print_matrix(matU);fprintf(stderr, "transposed matrix of right singular values:\n");print_matrix(matVt);fprintf(stderr, "\nopencv singular value decomposition:\n");cv::Mat mat(rows, cols, CV_32FC1);for (int y = 0; y < rows; ++y) {for (int x = 0; x < cols; ++x) {mat.at<float>(y, x) = vec.at(y).at(x);}}/*w calculated singular valuesu calculated left singular vectorsvt transposed matrix of right singular vectors*/cv::Mat w, u, vt, v;cv::SVD::compute(mat, w, u, vt, 4);//cv::transpose(vt, v);fprintf(stderr, "singular values:\n");print_matrix(w);fprintf(stderr, "left singular vectors:\n");print_matrix(u);fprintf(stderr, "transposed matrix of right singular values:\n");print_matrix(vt);return 0;
}

执行结果如下：

以下是采用Eigen实现的矩阵奇异值分解code：

#include "funset.hpp"
#include <math.h>
#include <iostream>
#include <vector>
#include <string>
#include <opencv2/opencv.hpp>
#include <Eigen/Dense>
#include "common.hpp"int test_SVD()
{//std::vector<std::vector<float>> vec{ { 1.2f, 2.5f, 5.6f, -2.5f },//				{ -3.6f, 9.2f, 0.5f, 7.2f },//				{ 4.3f, 1.3f, 9.4f, -3.4f },//				{ 6.4f, 0.1f, -3.7f, 0.9f } };//const int rows{ 4 }, cols{ 4 };//std::vector<std::vector<float>> vec{ { 1.2f, 2.5f, 5.6f, -2.5f },//				{ -3.6f, 9.2f, 0.5f, 7.2f },//				{ 4.3f, 1.3f, 9.4f, -3.4f } };//const int rows{ 3 }, cols{ 4 };std::vector<std::vector<float>> vec{ { 0.68f, 0.597f },{ -0.211f, 0.823f },{ 0.566f, -0.605f } };const int rows{ 3 }, cols{ 2 };std::vector<float> vec_;for (int i = 0; i < rows; ++i) {vec_.insert(vec_.begin() + i * cols, vec[i].begin(), vec[i].end());}Eigen::Map<Eigen::Matrix<float, Eigen::Dynamic, Eigen::Dynamic, Eigen::RowMajor>> m(vec_.data(), rows, cols);fprintf(stderr, "source matrix:\n");std::cout << m << std::endl;Eigen::JacobiSVD<Eigen::MatrixXf> svd(m, Eigen::ComputeFullV | Eigen::ComputeFullU); // ComputeThinU | ComputeThinVEigen::MatrixXf singular_values = svd.singularValues();Eigen::MatrixXf left_singular_vectors = svd.matrixU();Eigen::MatrixXf right_singular_vectors = svd.matrixV();fprintf(stderr, "singular values:\n");print_matrix(singular_values.data(), singular_values.rows(), singular_values.cols());fprintf(stderr, "left singular vectors:\n");print_matrix(left_singular_vectors.data(), left_singular_vectors.rows(), left_singular_vectors.cols());fprintf(stderr, "right singular vecotrs:\n");print_matrix(right_singular_vectors.data(), right_singular_vectors.rows(), right_singular_vectors.cols());return 0;
}

执行结果如下：

由以上结果可见：C++、OpenCV、Eigen结果是相同的。

GitHub：

https://github.com/fengbingchun/NN_Test

https://github.com/fengbingchun/Eigen_Test