矩阵伪逆介绍及C++/OpenCV/Eigen的三种实现

矩阵A的伪逆(Moore-Penrose pseudoinverse)定义为：A⁺=VD⁺U^T,其中，U，D和V是矩阵A奇异值分解后得到的矩阵。对角矩阵D的伪逆D⁺是非零元素取倒数之后再转置得到的。

当矩阵A的列数多于行数时，使用伪逆求解线性方程是众多可能解法中的一种。特别地，x=A⁺y是方程所有可行解中欧几里得范数‖x‖₂最小的一个。

当矩阵A的行数多于列数时，可能没有解。在这种情况下，通过伪逆得到的x使得Ax和y的欧几里得距离‖Ax-y‖₂最小。

广义逆阵(generalized inverse)也称为伪逆矩阵(pseudoinverse)，是在数学矩阵领域内的名词，一矩阵A的广义逆阵是指另一矩阵具有部分逆矩阵的特性，但是不一定具有逆矩阵的所有特性。假设一矩阵A∈R^n*m及另一矩阵A^g∈^m*n，若A^g满足此条件，AA^gA=A，则A^g即为A的逆矩阵。构建广义逆阵的目的是针对可逆矩阵以外的矩阵(例如非方阵的矩阵)可以找到一矩阵有一些类似逆矩阵的特性。任意的矩阵都存在广义逆阵，若一矩阵存在逆矩阵，逆矩阵即为其唯一的广义逆阵。有些广义逆阵可以定义在和结合律乘法有关的数学结构中。可以借助SVD(奇异值分解)来求解伪逆。

以上内容摘自：《深度学习中文版》和维基百科

以下是分别采用C++和OpenCV实现的求矩阵伪逆：

#include "funset.hpp"
#include <math.h>
#include <iostream>
#include <string>
#include <vector>
#include <opencv2/opencv.hpp>
#include "common.hpp"

// ================================= 求伪逆矩阵 =================================
template<typename _Tp>
int pinv(const std::vector<std::vector<_Tp>>& src, std::vector<std::vector<_Tp>>& dst, _Tp tolerance)
{
	std::vector<std::vector<_Tp>> D, U, Vt;
	if (svd(src, D, U, Vt) != 0) {
		fprintf(stderr, "singular value decomposition fail\n");
		return -1;
	}

	int m = src.size();
	int n = src[0].size();

	std::vector<std::vector<_Tp>> Drecip, DrecipT, Ut, V;

	transpose(Vt, V);
	transpose(U, Ut);

	if (m < n)
		std::swap(m, n);

	Drecip.resize(n);
	for (int i = 0; i < n; ++i) {
		Drecip[i].resize(m, (_Tp)0);

		if (D[i][0] > tolerance)
			Drecip[i][i] = 1.0f / D[i][0];
	}

	if (src.size() < src[0].size())
		transpose(Drecip, DrecipT);
	else
		DrecipT = Drecip;

	std::vector<std::vector<_Tp>> tmp = matrix_mul(V, DrecipT);
	dst = matrix_mul(tmp, Ut);

	return 0;
}

template<typename _Tp> // mat1(m, n) * mat2(n, p) => result(m, p)
static std::vector<std::vector<_Tp>> matrix_mul(const std::vector<std::vector<_Tp>>& mat1, const std::vector<std::vector<_Tp>>& mat2)
{
	std::vector<std::vector<_Tp>> result;
	int m1 = mat1.size(), n1 = mat1[0].size();
	int m2 = mat2.size(), n2 = mat2[0].size();
	if (n1 != m2) {
		fprintf(stderr, "mat dimension dismatch\n");
		return result;
	}

	result.resize(m1);
	for (int i = 0; i < m1; ++i) {
		result[i].resize(n2, (_Tp)0);
	}

	for (int y = 0; y < m1; ++y) {
		for (int x = 0; x < n2; ++x) {
			for (int t = 0; t < n1; ++t) {
				result[y][x] += mat1[y][t] * mat2[t][x];
			}
		}
	}

	return result;
}

// ================================= 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; // urows
	std::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);
	else
		tmp_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_pseudoinverse()
{
	//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<std::vector<float>> vec{ { 0.68f, 0.597f, -0.211f },
						{ 0.823f, 0.566f, -0.605f } };
	const int rows{ 2 }, cols{ 3 };

	fprintf(stderr, "source matrix:\n");
	print_matrix(vec);

	fprintf(stderr, "\nc++ implement pseudoinverse:\n");
	std::vector<std::vector<float>> pinv1;
	float  pinvtoler = 1.e-6;
	if (pinv(vec, pinv1, pinvtoler) != 0) {
		fprintf(stderr, "C++ implement pseudoinverse fail\n");
		return -1;
	}
	print_matrix(pinv1);

	fprintf(stderr, "\nopencv implement pseudoinverse:\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);
		}
	}

	cv::Mat pinv2;
	cv::invert(mat, pinv2, cv::DECOMP_SVD);
	print_matrix(pinv2);

	return 0; 
}

执行结果如下：

以下是采用Eigen实现的求矩阵伪逆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_pseudoinverse()
{
	//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<std::vector<float>> vec{ { 0.68f, 0.597f, -0.211f },
					{ 0.823f, 0.566f, -0.605f } };
	const int rows{ 2 }, cols{ 3 };

	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;

	fprintf(stderr, "\nEigen implement pseudoinverse:\n");
	auto svd = m.jacobiSvd(Eigen::ComputeFullU | Eigen::ComputeFullV);

	const auto &singularValues = svd.singularValues();
	Eigen::Matrix<float, Eigen::Dynamic, Eigen::Dynamic> singularValuesInv(m.cols(), m.rows());
	singularValuesInv.setZero();
	double  pinvtoler = 1.e-6; // choose your tolerance wisely
	for (unsigned int i = 0; i < singularValues.size(); ++i) {
		if (singularValues(i) > pinvtoler)
			singularValuesInv(i, i) = 1.0f / singularValues(i);
		else
			singularValuesInv(i, i) = 0.f;
	}

	Eigen::MatrixXf pinvmat = svd.matrixV() * singularValuesInv * svd.matrixU().transpose();
	std::cout << pinvmat << std::endl;

	return 0;
}

执行结果如下：

由以上结果可见：C++、OpenCV、Eigen实现结果是一致的。

GitHub：

https://github.com/fengbingchun/NN_Test
https://github.com/fengbingchun/Eigen_Test

原文链接：https://blog.csdn.net/fengbingchun/article/details/72874623

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