Backends & GPU Support

The following is a brief overview of libraries which have been tested with opt_einsum:

  • tensorflow: compiled tensor expressions that can run on GPU.
  • theano: compiled tensor expressions that can run on GPU.
  • cupy: numpy-like api for GPU tensors.
  • dask: larger-than-memory tensor computations, distributed scheduling, and potential reuse of intermediaries.
  • sparse: sparse tensors.
  • pytorch: numpy-like api for GPU tensors.

opt_einsum is quite agnostic to the type of n-dimensional arrays (tensors) it uses, since finding the contraction path only relies on getting the shape attribute of each array supplied. It can perform the underlying tensor contractions with various libraries. In fact, any library that provides a tensordot() and transpose() implementation can perform most normal contractions. While more special functionality such as axes reduction is reliant on a einsum() implementation.


For a contraction to be possible without using a backend einsum, it must satisfy the following rule: in the full expression (so including output indices) each index must appear twice. In other words, each dimension must be contracted with one other dimension, or left alone.

General backend for any ndarray

This ‘duck-typing’ support just requires specifying the correct backend argument for the type of arrays supplied when calling contract(). For example, if you had a library installed called 'foo' which provided an ndarray like object with a .shape attribute as well as foo.tensordot and foo.transpose then you could contract then with something like:

contract(einsum_str, *foo_arrays, backend='foo')

Behind the scenes opt_einsum will find the contraction path, perform pairwise contractions using e.g. foo.tensordot and finally return whatever type those functions return.


dask is an example of a library which satisfies these requirements. For example:

>>> import opt_einsum as oe
>>> import dask.array as da
>>> shapes = (3, 200), (200, 300), (300, 4)
>>> dxs = [da.random.normal(0, 1, shp, chunks=(100, 100)) for shp in shapes]
>>> dxs
[dask.array<da.random.normal, shape=(3, 200), dtype=float64, chunksize=(3, 100)>,
 dask.array<da.random.normal, shape=(200, 300), dtype=float64, chunksize=(100, 100)>,
 dask.array<da.random.normal, shape=(300, 4), dtype=float64, chunksize=(100, 4)>]

>>> dy = oe.contract("ab,bc,cd", *dxs, backend='dask')
>>> dy
dask.array<transpose, shape=(3, 4), dtype=float64, chunksize=(3, 4)>

>>> dy.compute()
array([[ 470.71404665,    2.44931372,  -28.47577265,  424.37716615],
       [  64.38328345, -287.40753131,  144.46515642,  324.88169821],
       [-142.07153553, -180.41739259,  125.0973783 , -239.16754541]])

In this case, dask arrays in = dask array out, since dask arrays have a shape attribute, and opt_einsum can find dask.array.tensordot and dask.array.transpose.


The sparse library also fits the bill and is supported. An example:

>>> import opt_einsum as oe
>>> import sparse as sp
>>> shapes = (3, 200), (200, 300), (300, 4)
>>> sxs = [sp.random(shp) for shp in shapes]
[<COO: shape=(3, 200), dtype=float64, nnz=6, sorted=False, duplicates=True>,
 <COO: shape=(200, 300), dtype=float64, nnz=600, sorted=False, duplicates=True>,
 <COO: shape=(300, 4), dtype=float64, nnz=12, sorted=False, duplicates=True>]

>>> sy = oe.contract("ab,bc,cd", *sxs, backend='sparse')
<COO: shape=(3, 4), dtype=float64, nnz=0, sorted=False, duplicates=False>

Special (GPU) backends for numpy arrays

A special case is if you want to supply numpy arrays and get numpy arrays back, but use a different backend, such as performing a contraction on a GPU. Unless the specified backend works on numpy arrays this requires converting to and from the backend array type. Currently opt_einsum can handle this automatically for:

which all offer GPU support. Since tensorflow and theano both require compiling the expression, this functionality is encapsulated in generating a ContractExpression using contract_expression(), which can then be called using numpy arrays whilst specifiying backend='tensorflow' etc. Additionally, if arrays are marked as constant (see Specifying Constants), then these arrays will be kept on the device for optimal performance.


If theano is installed, using it as backend is as simple as specifiying backend='theano':

>>> import opt_einsum as oe
>>> shapes = (3, 200), (200, 300), (300, 4)
>>> expr = oe.contract_expression("ab,bc,cd", *shapes)
>>> expr

>>> import numpy as np
>>> # GPU advantage mainly for low precision numbers
>>> xs = [np.random.randn(*shp).astype(np.float32) for shp in shapes]
>>> expr(*xs, backend='theano')  # might see some fluff on first run
array([[ 129.28352  , -128.00702  , -164.62917  , -335.11682  ],
       [-462.52344  , -121.12657  ,  -67.847626 ,  624.5457   ],
       [   5.2838974,   36.441578 ,   81.62851  ,  703.1576   ]],

Note that you can still supply theano.tensor.TensorType directly to opt_einsum (with backend='theano'), and it will return the relevant theano type.


To run the expression with tensorflow, you need to register a default session:

>>> import tensorflow as tf
>>> sess = tf.Session()  # might see some fluff

>>> with sess.as_default(): out = expr(*xs, backend='tensorflow')
>>> out
array([[ 129.28357  , -128.00684  , -164.62903  , -335.1167   ],
       [-462.52362  , -121.12659  ,  -67.84769  ,  624.5455   ],
       [   5.2839584,   36.44155  ,   81.62852  ,  703.15784  ]],

Note that you can still supply this expression with, for example, a tensorflow.placeholder using backend='tensorflow', and then no conversion would take place, instead you’d get a tensorflow.Tensor back.

Version 1.9 of tensorflow also added support for eager execution of computations. If compilation of the contraction expression tensorflow graph is taking a substantial amount of time up then it can be advantageous to use this, especially since tensor contractions are quite compute-bound. This is achieved by running the following snippet:

import tensorflow as tf

After which opt_einsum will automatically detect eager mode if backend='tensorflow' is supplied to a ContractExpression.

Pytorch & Cupy

Both pytorch and cupy offer numpy-like, GPU-enabled arrays which execute eagerly rather than requiring any compilation. If they are installed, no steps are required to utilize them other than specifiying the backend keyword:

>>> expr(*xs, backend='torch')
array([[ 129.28357  , -128.00684  , -164.62903  , -335.1167   ],
       [-462.52362  , -121.12659  ,  -67.84769  ,  624.5455   ],
       [   5.2839584,   36.44155  ,   81.62852  ,  703.15784  ]],

>>> expr(*xs, backend='cupy')
array([[ 129.28357  , -128.00684  , -164.62903  , -335.1167   ],
       [-462.52362  , -121.12659  ,  -67.84769  ,  624.5455   ],
       [   5.2839584,   36.44155  ,   81.62852  ,  703.15784  ]],

And as with the other GPU backends, if raw cupy or pytorch arrays are supplied the returned array will be of the same type, with no conversion to or from numpy arrays.