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Optimizing numpys einsum function

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News: Opt_einsum will be in NumPy 1.12 and BLAS features in NumPy 1.14! Call opt_einsum as np.einsum(..., optimize=True). This repository contains more advanced features such as Dask or Tensorflow backends as well as a testing ground for newer features in this ecosystem.

Optimized Einsum: A tensor contraction order optimizer

Optimized einsum can greatly reduce the overall time np.einsum takes by optimizing the expression's contraction order and dispatching many operations to canonical BLAS routines. See the documention for more information.

As well as opt_einsum.contract acting as a drop-in replacement for np.einsum, the following capabilities are enabled by opt_einsum:

Quick tutorial

Einsum is a 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, let us examine the following index transformation: M_{pqrs} = C_{pi} C_{qj} I_{ijkl} C_{rk} C_{sl}

We can then develop two seperate implementations that produce the same result:

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

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

%timeit naive(I, C)
1 loops, best of 3: 934 ms per loop

%timeit optimized(I, C)
1000 loops, best of 3: 527 us per loop

A 2000 fold speed up for 4 extra lines of code! 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 significantly. 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 this logic for you:

from opt_einsum import contract

%timeit contract('pi,qj,ijkl,rk,sl->pqrs', C, C, I, C, C)
1000 loops, best of 3: 324 us 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 under the hood to exploit any vendor BLAS functionality that your NumPy build has!

Please see the documentation for more features!


opt_einsum can either be installed via pip install opt_einsum or from conda conda install opt_einsum -c conda-forge. See the installation documenation for further methods.


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



All contributions, bug reports, bug fixes, documentation improvements, enhancements, and ideas are welcome.

A detailed overview on how to contribute can be found in the contributing guide.

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