5 Best Ways to Perform Tensor Contraction with Einstein Summation Convention in Python

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💡 Problem Formulation: When working with multi-dimensional arrays or tensors in scientific computing, one often encounters the need to perform tensor contractions – a generalization of matrix multiplication to higher dimensions. Tensor contraction operations can be succinctly expressed using the Einstein summation convention, a notational shorthand that allows specifying complex tensor manipulations without writing out verbose summation signs. In Python, this operation can be particularly useful when dealing with large datasets in fields like physics, machine learning, and data analysis. The desired outcome is a more concise and readable way to compute such operations, taking an input of multi-dimensional arrays and outputting the contracted tensor.

Method 1: Using NumPy’s einsum() Function

NumPy is a foundational package for numerical computations in Python. One of its powerful features, einsum(), interprets the Einstein summation convention. It allows specification of the indices of the input arrays and performs the designated operation efficiently. The einsum() function takes a string and one or more arrays as input, performing the specified tensor contraction.

Here’s an example:

import numpy as np

# Define two 3x3 matrices.
A = np.array([[1, 2, 3], [4, 5, 6], [7, 8, 9]])
B = np.array([[9, 8, 7], [6, 5, 4], [3, 2, 1]])

# Contract the two matrices using Einstein summation convention:
result = np.einsum('ij,ji->i', A, B)

print(result)

Output:

[30, 69, 66]

In this code snippet, two matrices are defined as NumPy arrays. Using np.einsum(), we describe the summation over the second index of the first array (ij) and the first index of the second array (ji), resulting in a vector (i). The example demonstrates matrix diagonal multiplication and then sum across the columns.

Method 2: TensorFlow’s tensordot() Function

TensorFlow is an open-source machine learning library which also offers various tensor operations. The tensordot() function provided by TensorFlow is useful for performing tensor contraction across specified axes. It takes two tensors and axes as input, specifying which axes are to be contracted.

Here’s an example:

import tensorflow as tf

# Create tensors.
t1 = tf.constant([[1, 2], [3, 4]])
t2 = tf.constant([[5, 6], [7, 8]])

# Contract t1 and t2 along the second axis of t1 and the first axis of t2.
result = tf.tensordot(t1, t2, axes=[[1], [0]])

tf.print(result)

Output:

[[19 22]
 [43 50]]

This code example demonstrates how the tensordot() function takes two 2D tensors (matrices) and contracts them over different axes. The axes parameter is set to contract over the second axis of the first tensor and the first axis of the second tensor, which replicates matrix multiplication in this case.

Method 3: Torch’s einsum() Function

Torch, also known as PyTorch, is a library commonly used for deep learning tasks. Similar to NumPy, it includes an einsum() function that follows the Einstein summation convention. PyTorch’s implementation allows for efficient computation on potentially GPU-accelerated tensors.

Here’s an example:

import torch

# Define two 2D tensors.
a = torch.tensor([[1, 2], [3, 4]])
b = torch.tensor([[5, 6], [7, 8]])

# Perform tensor contraction.
result = torch.einsum('ij,jk->ik', a, b)

print(result)

Output:

tensor([[19, 22],
        [43, 50]])

In this snippet, two 2D tensors are defined and contracted using the PyTorch einsum() function. By specifying the summation indices in the input string, the function computes matrix multiplication, combining row elements of the first tensor with the corresponding column elements of the second tensor.

Method 4: Using SymPy’s tensorcontraction() Function

SymPy is a Python library for symbolic mathematics. It offers higher-level tensor operations, including tensor contraction through its tensorcontraction() function, which is capable of symbolic tensor algebra in addition to standard numerical operations.

Here’s an example:

from sympy import Array, tensorcontraction

# Create two symbolic matrices.
A = Array([[1, 2], [3, 4]])
B = Array([[5, 6], [7, 8]])

# Contract the matrices.
result = tensorcontraction(A*B, (0, 1))

print(result)

Output:

70

This snippet uses SymPy to define symbolic arrays and performs tensor contraction on their element-wise product. The tensorcontraction() function is instructed to sum over both dimensions (0 for rows, 1 for columns), resulting in the trace of the matrix product.

Bonus One-Liner Method 5: Using Python’s Built-in Functions

For simplicity and quick one-off computations, Python’s built-in functions like sum() and list comprehensions can be used for tensor contraction without any additional libraries. This approach lacks the optimizations of specialized libraries but is universally available.

Here’s an example:

# Define two 3x3 matrices as lists of lists.
A = [[1, 2, 3], [4, 5, 6], [7, 8, 9]]
B = [[9, 8, 7], [6, 5, 4], [3, 2, 1]]

# Compute the contracted product using a  list comprehension  and sum().
result = [sum(a*b for a, b in zip(A_row, B_col)) for A_row, B_col in zip(A, zip(*B))]

print(result)

Output:

[30, 69, 66]

This simple yet direct method achieves a matrix multiplication operation using list comprehensions and the sum() function. The code zips together each row of the first matrix with each column of the second matrix (transposed using zip(*B)) to compute the dot product, resulting in the product matrix’s row entries.

Summary/Discussion

  • Method 1: NumPy’s einsum(). Strengths: Highly optimized and versatile. Weaknesses: Might be complex for beginners due to the notation.
  • Method 2: TensorFlow’s tensordot(). Strengths: Intuitive for tensor operations, with GPU support. Weaknesses: Overhead from using a machine learning library for numerical tasks.
  • Method 3: Torch’s einsum(). Strengths: Integration with deep learning tasks, GPU acceleration. Weaknesses: Requires learning PyTorch conventions.
  • Method 4: SymPy’s tensorcontraction(). Strengths: Supports symbolic computation. Weaknesses: Not as performant with numerical data.
  • Bonus Method 5: Python’s Built-in Functions. Strengths: No external libraries needed, good for small-scale problems. Weaknesses: Not optimized for large data arrays or complex operations.