einsum() is a very powerful function for contracting tensors of arbitrary dimension and index. However, it is only optimized to contract two terms at a time resulting in non-optimal scaling.

For example, consider the following index transformation: M_{pqrs} = C_{pi} C_{qj} I_{ijkl} C_{rk} C_{sl}

Consider two different algorithms:

import numpy as np

dim = 10
I = np.random.rand(dim, dim, dim, dim)
C = np.random.rand(dim, dim)

def naive(I, C):
    # N^8 scaling
    return np.einsum('pi,qj,ijkl,rk,sl->pqrs', C, C, I, C, C)

def optimized(I, C):
    # N^5 scaling
    K = np.einsum('pi,ijkl->pjkl', C, I)
    K = np.einsum('qj,pjkl->pqkl', C, K)
    K = np.einsum('rk,pqkl->pqrl', C, K)
    K = np.einsum('sl,pqrl->pqrs', C, K)
    return K

The einsum function does not consider building intermediate arrays; therefore, helping einsum out by building these intermediate arrays can result in a considerable cost savings even for small N (N=10):

>> np.allclose(naive(I, C), optimized(I, C))

%timeit naive(I, C)
1 loops, best of 3: 829 ms per loop

%timeit optimized(I, C)
1000 loops, best of 3: 445 µs per loop

The index transformation is a well known contraction that leads to straightforward intermediates. This contraction can be further complicated by considering that the shape of the C matrices need not be the same, in this case the ordering in which the indices are transformed matters greatly. Logic can be built that optimizes the ordering; however, this is a lot of time and effort for a single expression.

The opt_einsum package is a drop in replacement for the np.einsum function and can handle all of the logic for you:

from opt_einsum import contract

dim = 30
I = np.random.rand(dim, dim, dim, dim)
C = np.random.rand(dim, dim)

%timeit optimized(I, C)
10 loops, best of 3: 65.8 ms per loop

%timeit contract('pi,qj,ijkl,rk,sl->pqrs', C, C, I, C, C)
100 loops, best of 3: 16.2 ms per loop

The above will automatically find the optimal contraction order, in this case identical to that of the optimized function above, and compute the products for you. In this case, it even uses np.dot under the hood to exploit any vendor BLAS functionality that your NumPy build has!

We can then view more details about the optimized contraction order:

>>> from opt_einsum import contract_path

>>> path_info = oe.contract_path('pi,qj,ijkl,rk,sl->pqrs', C, C, I, C, C)

>>> print(path_info[0])
[(0, 2), (0, 3), (0, 2), (0, 1)]

>>> print(path_info[1])
  Complete contraction:  pi,qj,ijkl,rk,sl->pqrs
         Naive scaling:  8
     Optimized scaling:  5
      Naive FLOP count:  8.000e+08
  Optimized FLOP count:  8.000e+05
   Theoretical speedup:  1000.000
  Largest intermediate:  1.000e+04 elements
scaling   BLAS                  current                                remaining
   5      GEMM            ijkl,pi->jklp                      qj,rk,sl,jklp->pqrs
   5      GEMM            jklp,qj->klpq                         rk,sl,klpq->pqrs
   5      GEMM            klpq,rk->lpqr                            sl,lpqr->pqrs
   5      GEMM            lpqr,sl->pqrs                               pqrs->pqrs


If this code has benefited your research, please support us by citing:

Daniel G. A. Smith and Johnnie Gray, opt_einsum - A Python package for optimizing contraction order for einsum-like expressions. Journal of Open Source Software, 2018, 3(26), 753

DOI: https://doi.org/10.21105/joss.00753