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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 N = 10 C = np.random.rand(N, N) I = np.random.rand(N, N, N, N) 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)) True %timeit naive(I, C) 1 loops, best of 3: 1.18 s per loop %timeit optimized(I, C) 1000 loops, best of 3: 612 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 contract('pi,qj,ijkl,rk,sl->pqrs', C, C, I, C, C)

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

TODO: Figure out how to actually get changelog content.

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