了解 PyTorch einsum

Question

我很熟悉einsum works in NumPy. A similar functionality is also offered by PyTorch: torch.einsum()。在功能或性能方面有何异同？ PyTorch 文档中提供的信息相当少，没有提供任何关于此的见解。

Answer 1

由于火炬文档中对 einsum 的描述很简陋，我决定写这个 post 来记录、比较和对比如何 torch.einsum() behaves when compared to numpy.einsum()。

差异：

NumPy 允许“下标字符串”的小写字母和大写字母 [a-zA-Z] 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（或元组元组、元组列表、列表元组）甚至 PyTorch 张量作为操作数（即输入）。这是因为 操作数 只需 array_like 而不是严格的 NumPy nd 数组。相反，PyTorch 期望 operands（即输入）严格是 PyTorch 张量。如果您传递纯 Python lists/tuples（或其组合）或 NumPy nd 数组，它将抛出 TypeError。
除了 nd-arrays 之外，NumPy 还支持很多关键字参数（例如 optimize），而 PyTorch 还没有提供这种灵活性。

以下是一些示例在 PyTorch 和 NumPy 中的实现：

# 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
火炬：torch.matmul(aten, bten)； aten.mm(bten)
麻木：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
火炬：torch.diag(aten)
麻木：np.einsum("ii -> i", arr)

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

3) Hadamard 乘积（即两个张量的逐元素乘积）
火炬：aten * bten
麻木：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) 逐元素平方
火炬：aten ** 2
麻木：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: Element-wise nth power 可以通过重复下标串和tensor n 次。例如，计算张量的逐元素四次方可以使用：

# 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) 迹线（即主对角线元素之和）
火炬：torch.trace(aten)
NumPy einsum：np.einsum("ii -> ", arr)

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

6) 矩阵转置
火炬：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）
火炬：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) 内积（向量） 火炬：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) 沿轴求和 0
火炬：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
火炬：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
火炬：torch.bmm(batch_tensor_1, batch_tensor_2)
麻木：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 求和
火炬：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) 对nD张量中的所有元素求和
火炬：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（即边缘化）
火炬：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) 双点积/Frobenius inner product（等同于：torch.sum（hadamard-product）cf. 3）
火炬：torch.sum(aten * bten)
麻木：np.einsum("ij, ij -> ", arr1, arr2)

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

了解 PyTorch einsum

Understanding PyTorch einsum

python

numpy

numpy-einsum

pytorch

tensor