How to utilize symmetry in tensor contraction

When working with tensors, one common problem is how to efficiently perform tensor contractions while taking advantage of any available symmetry. In this article, we will explore three different approaches to solve this problem in Julia.

Approach 1: Manual Symmetry Exploitation

In this approach, we manually exploit the symmetry of the tensors to reduce the number of computations required for the contraction. We can achieve this by rearranging the indices of the tensors to match their symmetries before performing the contraction.

# Julia code for manual symmetry exploitation
function tensor_contraction(A, B)
    # Rearrange indices of A and B to match their symmetries
    A_sym = permutedims(A, (2, 1, 3))
    B_sym = permutedims(B, (1, 3, 2))
    
    # Perform tensor contraction
    C = A_sym * B_sym
    
    return C
end

This approach requires manual identification and manipulation of the tensor symmetries, which can be time-consuming and error-prone. However, it can lead to significant computational savings if the symmetries are properly exploited.

Approach 2: SymPy.jl Library

SymPy.jl is a Julia library that provides symbolic computation capabilities. It can be used to automatically detect and exploit tensor symmetries during the contraction process.

# Julia code using SymPy.jl
using SymPy

function tensor_contraction(A, B)
    # Convert A and B to symbolic tensors
    A_sym = Sym(A)
    B_sym = Sym(B)
    
    # Perform tensor contraction
    C_sym = A_sym * B_sym
    
    # Convert C_sym back to a Julia array
    C = Array(C_sym)
    
    return C
end

This approach leverages the symbolic computation capabilities of SymPy.jl to automatically detect and exploit tensor symmetries. It eliminates the need for manual identification and manipulation of symmetries, making it more convenient and less error-prone.

Approach 3: TensorOperations.jl Library

TensorOperations.jl is a Julia library specifically designed for efficient tensor contractions. It provides a high-level interface for expressing tensor contractions and automatically optimizes the computation based on the available symmetries.

# Julia code using TensorOperations.jl
using TensorOperations

function tensor_contraction(A, B)
    # Define the contraction expression
    @tensor C[i, j, k] := A[i, l, m] * B[l, m, j]
    
    return C
end

This approach leverages the high-level interface provided by TensorOperations.jl to express the tensor contraction in a concise and readable manner. The library automatically detects and exploits tensor symmetries, resulting in efficient computations.

Among the three options, Approach 3 using TensorOperations.jl is the recommended choice. It provides a high-level and optimized interface for tensor contractions, making it easier to express and compute tensor contractions while taking advantage of symmetry. It eliminates the need for manual identification and manipulation of symmetries, and it is specifically designed for efficient tensor computations.