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⚡️Optimizing einsum functions in NumPy, Tensorflow, Dask, and more with contraction order optimization.

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Optimized Einsum

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Optimized Einsum: A tensor contraction order optimizer

Optimized einsum can significantly reduce the overall execution time of einsum-like expressions (e.g.,

, ) by optimizing the expression's contraction order and dispatching many operations to canonical BLAS, cuBLAS, or other specialized routines.

Optimized einsum is agnostic to the backend and can handle NumPy, Dask, PyTorch, Tensorflow, CuPy, Sparse, Theano, JAX, and Autograd arrays as well as potentially any library which conforms to a standard API. See the documentation for more information.

Example usage


function can often act as a drop-in replacement for

functions without further changes to the code while providing superior performance. Here, a tensor contraction is preformed with and without optimization:
import numpy as np
from opt_einsum import contract

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

%timeit np.einsum('pi,qj,ijkl,rk,sl->pqrs', C, C, I, C, C) 1 loops, best of 3: 934 ms per loop

%timeit contract('pi,qj,ijkl,rk,sl->pqrs', C, C, I, C, C) 1000 loops, best of 3: 324 us per loop

In this particular example, we see a ~3000x performance improvement which is not uncommon when compared against unoptimized contractions. See the backend examples for more information on using other backends.


The algorithms found in this repository often power the

optimizations in many of the above projects. For example, the optimization of
has been passed upstream and most of the same features that can be found in this repository can be enabled with
np.einsum(..., optimize=True)
. However, this repository often has more up to date algorithms for complex contractions.

The following capabilities are enabled by


Please see the documentation for more features!


can either be installed via
pip install opt_einsum
or from conda
conda install opt_einsum -c conda-forge
. See the installation documentation 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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