# opt_einsum¶

`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))
True
%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
```

## Citation¶

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