Eigen学习笔记7：Reductions, visitors and broadcasting

Reductions

In Eigen, a reduction is a function taking a matrix or array, and returning a single scalar value.最常用的归约方法之一是.sum()，它返回给定矩阵或数组内所有元素的总和。

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#include <iostream>
#include <Eigen/Dense>

using namespace std;
int main()
{
Eigen::Matrix2d mat;
mat << 1, 2,
3, 4;
cout << "Here is mat.sum(): " << mat.sum() << endl;
cout << "Here is mat.prod(): " << mat.prod() << endl;
cout << "Here is mat.mean(): " << mat.mean() << endl;
cout << "Here is mat.minCoeff(): " << mat.minCoeff() << endl;
cout << "Here is mat.maxCoeff(): " << mat.maxCoeff() << endl;
cout << "Here is mat.trace(): " << mat.trace() << endl;
}

输出：

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ere is mat.sum(): 10
Here is mat.prod(): 24
Here is mat.mean(): 2.5
Here is mat.minCoeff(): 1
Here is mat.maxCoeff(): 4
Here is mat.trace(): 5

The trace of a matrix, as returned by the function trace(), is the sum of the diagonal coefficients and can equivalently be computed a.diagonal().sum().

范数计算

向量的平方范数可以通过squaredNorm()获得。它其系数的平方的绝对值之和。

Eigen还提供了norm()方法，该方法返回squaredNorm()的平方根。

这些运算也可以在矩阵上运算。在这种情况下，n×p矩阵被视为大小为（n * p）的向量，因此例如norm()方法返回“ Frobenius”或“ Hilbert-Schmidt”范数。

如果你想使用其他Lp范数，可以使用lpNorm< p >()方法。

p可以取Infinity，which is the maximum of the absolute values of the coefficients.

下面的示例演示了这些方法。

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#include <Eigen/Dense>
#include <iostream>

using namespace std;
using namespace Eigen;

int main()
{
VectorXf v(2);
MatrixXf m(2,2), n(2,2);

v << -1,
2;

m << 1,-2,
-3,4;

cout << "v.squaredNorm() = " << v.squaredNorm() << endl;
cout << "v.norm() = " << v.norm() << endl;
cout << "v.lpNorm<1>() = " << v.lpNorm<1>() << endl;
cout << "v.lpNorm<Infinity>() = " << v.lpNorm<Infinity>() << endl;

cout << endl;
cout << "m.squaredNorm() = " << m.squaredNorm() << endl;
cout << "m.norm() = " << m.norm() << endl;
cout << "m.lpNorm<1>() = " << m.lpNorm<1>() << endl;
cout << "m.lpNorm<Infinity>() = " << m.lpNorm<Infinity>() << endl;
}

输出：

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Operator norm: 1-norm和∞-norm可以通过其他方式得到。

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输出：

1 2	1-norm(m) = 6 == 6 infty-norm(m) = 7 == 7

有关这些表达式的语法的更多说明，请参见下文。

Boolean reductions

以下归约运算适用于布尔值：

all() returns true if all of the coefficients in a given Matrix or Array evaluate to true .
any() returns true if at least one of the coefficients in a given Matrix or Array evaluates to true .
count() returns the number of coefficients in a given Matrix or Array that evaluate to true.

这些通常与Array提供的按元素比较和相等运算符结合使用。

For instance, array > 0 is an Array of the same size as array , with true at those positions where the corresponding coefficient of array is positive. 因此，(array > 0).all()测试的所有系数是否array均为正。

在以下示例中可以看到：

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#include <Eigen/Dense>
#include <iostream>

using namespace std;
using namespace Eigen;

int main()
{
ArrayXXf a(2,2);

a << 1,2,
3,4;

cout << "(a > 0).all() = " << (a > 0).all() << endl;
cout << "(a > 0).any() = " << (a > 0).any() << endl;
cout << "(a > 0).count() = " << (a > 0).count() << endl;
cout << endl;
cout << "(a > 2).all() = " << (a > 2).all() << endl;
cout << "(a > 2).any() = " << (a > 2).any() << endl;
cout << "(a > 2).count() = " << (a > 2).count() << endl;
}

输出：

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(a > 0).all() = 1
(a > 0).any() = 1
(a > 0).count() = 4

(a > 2).all() = 0
(a > 2).any() = 1
(a > 2).count() = 2

用户定义的Reductions

查看DenseBase :: redux()函数。

Visitors

Visitors are useful when one wants to obtain the location of a coefficient inside a Matrix or Array.

最简单的示例是maxCoeff(＆x，＆y)和minCoeff(＆x，＆y)，用于查找Matrix或Array中最大或最小元素的位置。

传递给Visitors的参数是指向要存储行和列位置变量的指针。这些变量应为Index类型，如下所示：

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#include <iostream>
#include <Eigen/Dense>

using namespace std;
using namespace Eigen;

int main()
{
Eigen::MatrixXf m(2,2);

m << 1, 2,
3, 4;

//get location of maximum
MatrixXf::Index maxRow, maxCol;
float max = m.maxCoeff(&maxRow, &maxCol);

//get location of minimum
MatrixXf::Index minRow, minCol;
float min = m.minCoeff(&minRow, &minCol);

cout << "Max: " << max << ", at: " <<
maxRow << "," << maxCol << endl;
cout << "Min: " << min << ", at: " <<
minRow << "," << minCol << endl;
}

输出：

1 2	Max: 4, at: 1,1 Min: 1, at: 0,0

Partial reductions

Partial reductions是可以在Matrix或Array上按列或按行操作的reductions，对每个列或行应用reductions运算，并返回具有相应值的列或行向量。Partial reductions适用于colwise()或rowwise()。

一个简单的示例是获取给定矩阵中每一列中元素的最大值，并将结果存储在行向量中：

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#include <iostream>
#include <Eigen/Dense>

using namespace std;
int main()
{
Eigen::MatrixXf mat(2,4);
mat << 1, 2, 6, 9,
3, 1, 7, 2;

std::cout << "Column's maximum: " << std::endl
<< mat.colwise().maxCoeff() << std::endl;
}

输出：

1 2	Column's maximum: 3 2 7 9

可以逐行执行相同的操作：

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#include <iostream>
#include <Eigen/Dense>

using namespace std;
int main()
{
Eigen::MatrixXf mat(2,4);
mat << 1, 2, 6, 9,
3, 1, 7, 2;

std::cout << "Row's maximum: " << std::endl
<< mat.rowwise().maxCoeff() << std::endl;
}

输出：

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Row's maximum:
9
7

Note that column-wise operations return a row vector, while row-wise operations return a column vector.

将Partial reductions与其他操作结合起来

将Partial reductions的结果进行进一步处理。

Here is another example that finds the column whose sum of elements is the maximum within a matrix. With column-wise partial reductions this can be coded as:

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#include <iostream>
#include <Eigen/Dense>

using namespace std;
using namespace Eigen;
int main()
{
MatrixXf mat(2,4);
mat << 1, 2, 6, 9,
3, 1, 7, 2;

MatrixXf::Index maxIndex;
float maxNorm = mat.colwise().sum().maxCoeff(&maxIndex);

std::cout << "Maximum sum at position " << maxIndex << std::endl;

std::cout << "The corresponding vector is: " << std::endl;
std::cout << mat.col( maxIndex ) << std::endl;
std::cout << "And its sum is is: " << maxNorm << std::endl;
}

输出：

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Maximum sum at position 2
The corresponding vector is:
6
7
And its sum is is: 13

广播

广播背后的概念类似于 partial reductions，with the difference that broadcasting constructs an expression where a vector (column or row) is interpreted as a matrix by replicating it in one direction.

一个简单的示例是将某个列向量添加到矩阵中的每一列。这可以通过以下方式完成：

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#include <iostream>
#include <Eigen/Dense>

using namespace std;
int main()
{
Eigen::MatrixXf mat(2,4);
Eigen::VectorXf v(2);

mat << 1, 2, 6, 9,
3, 1, 7, 2;

v << 0,
1;

//add v to each column of m
mat.colwise() += v;

std::cout << "Broadcasting result: " << std::endl;
std::cout << mat << std::endl;
}

输出：

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Broadcasting result:
1 2 6 9
4 2 8 3

要指出的是，要逐列或逐行添加的向量必须为Vector类型，并且不能为Matrix。如果不满足，则会出现编译时错误。这也意味着，在使用Matrix进行操作时，只能对Vector类型的对象应用广播操作。这同样适用于Array类，其中等效VectorXf是ArrayXf。与往常一样，您不应在同一表达式中混合使用数组和矩阵。

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#include <iostream>
#include <Eigen/Dense>

using namespace std;
int main()
{
Eigen::MatrixXf mat(2,4);
Eigen::VectorXf v(4);

mat << 1, 2, 6, 9,
3, 1, 7, 2;

v << 0,1,2,3;

//add v to each row of m
mat.rowwise() += v.transpose();

std::cout << "Broadcasting result: " << std::endl;
std::cout << mat << std::endl;
}

输出：

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Broadcasting result:
1 3 8 12
3 2 9 5

将广播与其他操作结合

Broadcasting can also be combined with other operations, such as Matrix or Array operations, reductions and partial reductions.我们可以深入研究一个更高级的示例：

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#include <iostream>
#include <Eigen/Dense>

using namespace std;
using namespace Eigen;

int main()
{
Eigen::MatrixXf m(2,4);
Eigen::VectorXf v(2);

m << 1, 23, 6, 9,
3, 11, 7, 2;

v << 2,
3;

MatrixXf::Index index;
// find nearest neighbour
(m.colwise() - v).colwise().squaredNorm().minCoeff(&index);

cout << "Nearest neighbour is column " << index << ":" << endl;
cout << m.col(index) << endl;
}

输出：

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Nearest neighbour is column 0:
1
3