[Glean] Einsum Introduction
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Einsum is short for Einstein summation convention, which is a compact representation for combining products and sums in a general way. It is proposed by Einstein in 1916 to simplify tensor operations. In short, it is a shorthand notation for summation. It is a concise way to specify operations on multidimensional arrays. The syntax of einsum is as follows:
numpy.einsum(subscripts, *operands, out=None, dtype=None, order='K', casting='safe', optimize=False)
The subscripts argument is a string that specifies the subscripts for summation. The operands argument is a list of arrays to be operated on. The out argument is an optional output array. The dtype argument is an optional data type. The order argument is an optional memory layout. The casting argument is an optional casting rule. The optimize argument is an optional optimization flag.
Einsum could be used to replace but not limited to the following functions:
- Matrix trace: trace
- Matrix diagonal: diag
- Tensor summation (along axes): sum
- Tensor transpose: transpose
- Matrix multiplication: dot
- Tensor multiplication: tensordot
- Vector inner product: inner
- Outer product: outer
subscripts argument
The subscripts argument is a string that specifies the subscripts for summation. The subscripts consist of comma-separated subscript labels, where each label refers to a dimension of the corresponding operand. The labels are used to specify the dimensions of the operands to be multiplied and summed. The Einstein summation convention implies that a label that appears twice in the subscripts is to be summed over.
The detailed rules for the subscripts argument with examples are as follows:
- A label that appears only once in the subscripts is not summed over.
imeans thei-thdimension of the first operand.
- A label that appears twice in the subscripts is summed over.
ijmeans the sum of the product of thei-thdimension of the first operand and thej-thdimension of the second operand.
- Comma-separated labels in the subscripts imply a direct product.
i,jmeans the direct product of thei-thdimension of the first operand and thej-thdimension of the second operand.
- An ellipsis (
...) allows for an arbitrary number of dimensions.i...means all dimensions of the first operand except the first one.
->separates the input and output operands.ij,jk->ikmeans the sum of the product of thei-thdimension of the first operand and thej-thdimension of the second operand, and the result is stored in thek-thdimension of the output operand.
Example
import numpy as np
a = np.array([[1, 2], [3, 4]])
b = np.array([[5, 6], [7, 8]])
# Matrix multiplication
c = np.einsum('ij,jk->ik', a, b)
# c = np.dot(a, b)
# Matrix trace
d = np.einsum('ii', a)
# d = np.trace(a)
# Tensor summation
e = np.einsum('ijk->k', np.arange(24).reshape(2, 3, 4))
# e = np.sum(np.arange(24).reshape(2, 3, 4), axis=(0, 1))
# Tensor transpose
f = np.einsum('ijk->kji', np.arange(24).reshape(2, 3, 4))
# f = np.transpose(np.arange(24).reshape(2, 3, 4), axes=(2, 1, 0))
# Vector inner product
g = np.einsum('i,i', np.arange(3), np.arange(3))
# g = np.inner(np.arange(3), np.arange(3))
# Outer product
h = np.einsum('i,j->ij', np.arange(3), np.arange(3))
# h = np.outer(np.arange(3), np.arange(3))