[英]Multiply array of vectors with array of matrices; return array of vectors?
I've got a numpy
array of row vectors of shape (n,3) and another numpy array of matrices of shape (n,3,3). 我有一个
numpy
形状的行向量(N,3)和形状的矩阵的另一numpy的阵列的阵列(N,3,3)。 I would like to multiply each of the n vectors with the corresponding matrix and return an array of shape (n,3) of the resulting vectors. 我想将n个向量中的每一个与相应的矩阵相乘,并返回所得向量的形状(n,3)数组。
By now I've been using a for loop to iterate through the n vectors/matrices and do the multiplication item by item. 到目前为止,我一直在使用for循环遍历n个向量/矩阵,并逐项进行乘法运算。
I would like to know if there's a more numpy-ish way of doing this. 我想知道是否有更多的numpy-ish方式这样做。 A way without the for loop that might even be faster.
没有for循环的方式甚至可能更快。
//edit 1: //编辑1:
As requested, here's my loopy code (with n = 10
): 根据要求,这是我的循环代码(
n = 10
):
arr_in = np.random.randn(10, 3)
matrices = np.random.randn(10, 3, 3)
for i in range(arr_in.shape[0]): # 10 iterations
arr_out[i] = np.asarray(np.dot(arr_in[i], matrices[i]))
That dot-product
is essentially performing reduction along axis=1
of the two input arrays. 该
dot-product
基本上沿着两个输入阵列的axis=1
执行减少。 The dimensions could be represented like so - 尺寸可以表示如此 -
arr_in : n 3
matrices : n 3 3
So, one way to solve it would be to "push" the dimensions of arr_in
to front by one axis/dimension
, thus creating a singleton dimension at axis=2
in a 3D array version of it. 因此,解决它的一种方法是将
arr_in
的尺寸“推”到前面一个axis/dimension
,从而在它的3D阵列版本中在axis=2
处创建单个尺寸。 Then, sum-reducing the elements along axis = 1
would give us the desired output. 然后,沿
axis = 1
减少元素的总和将给出我们所需的输出。 Let's show it - 让我们展示一下 -
arr_in : n [3] 1
matrices : n [3] 3
Now, this could be achieved through two ways. 现在,这可以通过两种方式实现。
1) With np.einsum
- 1)使用
np.einsum
-
np.einsum('ij,ijk->ik',arr_in,matrices)
2) With NumPy broadcasting
- 2)借助
NumPy broadcasting
-
(arr_in[...,None]*matrices).sum(1)
Runtime test and verify output (for einsum
version) - 运行时测试和验证输出(对于
einsum
版本) -
In [329]: def loop_based(arr_in,matrices):
...: arr_out = np.zeros((arr_in.shape[0], 3))
...: for i in range(arr_in.shape[0]):
...: arr_out[i] = np.dot(arr_in[i], matrices[i])
...: return arr_out
...:
...: def einsum_based(arr_in,matrices):
...: return np.einsum('ij,ijk->ik',arr_in,matrices)
...:
In [330]: # Inputs
...: N = 16935
...: arr_in = np.random.randn(N, 3)
...: matrices = np.random.randn(N, 3, 3)
...:
In [331]: np.allclose(einsum_based(arr_in,matrices),loop_based(arr_in,matrices))
Out[331]: True
In [332]: %timeit loop_based(arr_in,matrices)
10 loops, best of 3: 49.1 ms per loop
In [333]: %timeit einsum_based(arr_in,matrices)
1000 loops, best of 3: 714 µs per loop
You could use np.einsum
. 你可以使用
np.einsum
。 To get v.dot(M)
for each vector-matrix pair, use np.einsum("...i,...ij", arr_in, matrices)
. 要获得每个向量 - 矩阵对的
v.dot(M)
,请使用np.einsum("...i,...ij", arr_in, matrices)
。 To get M.dot(v)
use np.einsum("...ij,...i", matrices, arr_in)
获取
M.dot(v)
使用np.einsum("...ij,...i", matrices, arr_in)
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