理解PyTorch einsum

Question

我熟悉合额在NumPy中的工作方式。PyTorch：torch.einsum()也提供了类似的功能。在功能或性能方面，有什么相似之处和不同之处？PyTorch文档中提供的信息非常少，没有提供任何有关这方面的见解。

Answer 1

由于einsum的描述在torch文档中比较少，所以我决定写这篇文章来记录、比较和对比torch.einsum()与numpy.einsum()的行为。

差异：

• NumPy允许小写字母和大写字母[a-zA-Z]用于“下标字符串”，而PyTorch只允许小写字母[a-z]。
• NumPy接受nd-数组、普通Python列表(或元组)、列表列表(或元组的元组、元组的列表、列表的元组)甚至PyTorch张量作为操作数(即输入)。这是因为操作数只能是array_like，而不是严格的NumPy和数组。相反，PyTorch期望操作数(即输入)严格地是PyTorch张量。如果您传递普通Python /tuple(或其组合)或NumPy nd-数组，它将抛出一个NumPy。
• NumPy除了支持nd-arrays之外，还支持很多关键字参数(例如optimize)，而PyTorch还没有提供这样的灵活性。

以下是PyTorch和NumPy中一些示例的实现：

# input tensors to work with

In [16]: vec
Out[16]: tensor([0, 1, 2, 3])

In [17]: aten
Out[17]: 
tensor([[11, 12, 13, 14],
        [21, 22, 23, 24],
        [31, 32, 33, 34],
        [41, 42, 43, 44]])

In [18]: bten
Out[18]: 
tensor([[1, 1, 1, 1],
        [2, 2, 2, 2],
        [3, 3, 3, 3],
        [4, 4, 4, 4]])

1)矩阵乘法

  PyTorch: `torch.matmul(aten, bten)` ; `aten.mm(bten)`

  NumPy : `np.einsum("ij, jk -> ik", arr1, arr2)` 

In [19]: torch.einsum('ij, jk -> ik', aten, bten)
Out[19]: 
tensor([[130, 130, 130, 130],
        [230, 230, 230, 230],
        [330, 330, 330, 330],
        [430, 430, 430, 430]])

2)沿主对角线提取元素。

PyTorch: `torch.diag(aten)`

NumPy : `np.einsum("ii -> i", arr)` 

In [28]: torch.einsum('ii -> i', aten)
Out[28]: tensor([11, 22, 33, 44])

3) Hadamard积(即两个张量的元素乘积)

PyTorch: `aten * bten`

NumPy : `np.einsum("ij, ij -> ij", arr1, arr2)` 

In [34]: torch.einsum('ij, ij -> ij', aten, bten)
Out[34]: 
tensor([[ 11,  12,  13,  14],
        [ 42,  44,  46,  48],
        [ 93,  96,  99, 102],
        [164, 168, 172, 176]])

4)元素向平方

PyTorch: `aten ** 2`

NumPy : `np.einsum("ij, ij -> ij", arr, arr)` 

In [37]: torch.einsum('ij, ij -> ij', aten, aten)
Out[37]: 
tensor([[ 121,  144,  169,  196],
        [ 441,  484,  529,  576],
        [ 961, 1024, 1089, 1156],
        [1681, 1764, 1849, 1936]])

General：元素级nth电源可以通过重复下标字符串和张量n时间来实现。例如，就计算单元而言，张量的第4次方可以使用以下方法完成：

# NumPy: np.einsum('ij, ij, ij, ij -> ij', arr, arr, arr, arr)
In [38]: torch.einsum('ij, ij, ij, ij -> ij', aten, aten, aten, aten)
Out[38]: 
tensor([[  14641,   20736,   28561,   38416],
        [ 194481,  234256,  279841,  331776],
        [ 923521, 1048576, 1185921, 1336336],
        [2825761, 3111696, 3418801, 3748096]])

5)迹(即主对角线元素之和)

PyTorch: `torch.trace(aten)`

NumPy einsum: `np.einsum("ii -> ", arr)` 

In [44]: torch.einsum('ii -> ', aten)
Out[44]: tensor(110)

6)矩阵转置

PyTorch: `torch.transpose(aten, 1, 0)`

NumPy einsum: `np.einsum("ij -> ji", arr)` 

In [58]: torch.einsum('ij -> ji', aten)
Out[58]: 
tensor([[11, 21, 31, 41],
        [12, 22, 32, 42],
        [13, 23, 33, 43],
        [14, 24, 34, 44]])

7)外积(向量)

PyTorch: `torch.ger(vec, vec)`

NumPy einsum: `np.einsum("i, j -> ij", vec, vec)` 

In [73]: torch.einsum('i, j -> ij', vec, vec)
Out[73]: 
tensor([[0, 0, 0, 0],
        [0, 1, 2, 3],
        [0, 2, 4, 6],
        [0, 3, 6, 9]])

8) (向量的内积) PyTorch：torch.dot(vec1, vec2)

NumPy einsum: `np.einsum("i, i -> ", vec1, vec2)` 

In [76]: torch.einsum('i, i -> ', vec, vec)
Out[76]: tensor(14)

9)沿轴0的求和

PyTorch: `torch.sum(aten, 0)`

NumPy einsum: `np.einsum("ij -> j", arr)` 

In [85]: torch.einsum('ij -> j', aten)
Out[85]: tensor([104, 108, 112, 116])

10)沿轴1求和

 PyTorch: `torch.sum(aten, 1)`

 NumPy einsum: `np.einsum("ij -> i", arr)` 

In [86]: torch.einsum('ij -> i', aten)
Out[86]: tensor([ 50,  90, 130, 170])

11)批量矩阵乘法

 PyTorch: `torch.bmm(batch_tensor_1, batch_tensor_2)`

 NumPy  : `np.einsum("bij, bjk -> bik", batch_tensor_1, batch_tensor_2)` 

# input batch tensors to work with
In [13]: batch_tensor_1 = torch.arange(2 * 4 * 3).reshape(2, 4, 3)
In [14]: batch_tensor_2 = torch.arange(2 * 3 * 4).reshape(2, 3, 4) 

In [15]: torch.bmm(batch_tensor_1, batch_tensor_2)  
Out[15]: 
tensor([[[  20,   23,   26,   29],
         [  56,   68,   80,   92],
         [  92,  113,  134,  155],
         [ 128,  158,  188,  218]],

        [[ 632,  671,  710,  749],
         [ 776,  824,  872,  920],
         [ 920,  977, 1034, 1091],
         [1064, 1130, 1196, 1262]]])

# sanity check with the shapes
In [16]: torch.bmm(batch_tensor_1, batch_tensor_2).shape 
Out[16]: torch.Size([2, 4, 4])

# batch matrix multiply using einsum
In [17]: torch.einsum("bij, bjk -> bik", batch_tensor_1, batch_tensor_2)
Out[17]: 
tensor([[[  20,   23,   26,   29],
         [  56,   68,   80,   92],
         [  92,  113,  134,  155],
         [ 128,  158,  188,  218]],

        [[ 632,  671,  710,  749],
         [ 776,  824,  872,  920],
         [ 920,  977, 1034, 1091],
         [1064, 1130, 1196, 1262]]])

# sanity check with the shapes
In [18]: torch.einsum("bij, bjk -> bik", batch_tensor_1, batch_tensor_2).shape

12)沿轴2的和

 PyTorch: `torch.sum(batch_ten, 2)`

 NumPy einsum: `np.einsum("ijk -> ij", arr3D)` 

In [99]: torch.einsum("ijk -> ij", batch_ten)
Out[99]: 
tensor([[ 50,  90, 130, 170],
        [  4,   8,  12,  16]])

13)将nD张量中的所有元素相加

 PyTorch: `torch.sum(batch_ten)`

 NumPy einsum: `np.einsum("ijk -> ", arr3D)` 

In [101]: torch.einsum("ijk -> ", batch_ten)
Out[101]: tensor(480)

14)多轴求和(即边际化)

 PyTorch: `torch.sum(arr, dim=(dim0, dim1, dim2, dim3, dim4, dim6, dim7))`

 NumPy: `np.einsum("ijklmnop -> n", nDarr)` 

# 8D tensor
In [103]: nDten = torch.randn((3,5,4,6,8,2,7,9))
In [104]: nDten.shape
Out[104]: torch.Size([3, 5, 4, 6, 8, 2, 7, 9])

# marginalize out dimension 5 (i.e. "n" here)
In [111]: esum = torch.einsum("ijklmnop -> n", nDten)
In [112]: esum
Out[112]: tensor([  98.6921, -206.0575])

# marginalize out axis 5 (i.e. sum over rest of the axes)
In [113]: tsum = torch.sum(nDten, dim=(0, 1, 2, 3, 4, 6, 7))

In [115]: torch.allclose(tsum, esum)
Out[115]: True

15)双点产品/ Frobenius内积 (同:torch.sum(hadamard- Products )) cf.3)

 PyTorch: `torch.sum(aten * bten)`

 NumPy  : `np.einsum("ij, ij -> ", arr1, arr2)` 

In [120]: torch.einsum("ij, ij -> ", aten, bten)
Out[120]: tensor(1300)