numpy.einsum_path#

numpy.einsum_path(subscripts,*operands,optimize='greedy')[source]#

Evaluates the lowest cost contraction order for an einsum expression byconsidering the creation of intermediate arrays.

Parameters:
subscriptsstr

Specifies the subscripts for summation.

*operandslist of array_like

These are the arrays for the operation.

optimize{bool, list, tuple, ‘greedy’, ‘optimal’}

Choose the type of path. If a tuple is provided, the second argument isassumed to be the maximum intermediate size created. If only a singleargument is provided the largest input or output array size is usedas a maximum intermediate size.

  • if a list is given that starts witheinsum_path, uses this as thecontraction path

  • if False no optimization is taken

  • if True defaults to the ‘greedy’ algorithm

  • ‘optimal’ An algorithm that combinatorially explores all possibleways of contracting the listed tensors and chooses the least costlypath. Scales exponentially with the number of terms in thecontraction.

  • ‘greedy’ An algorithm that chooses the best pair contractionat each step. Effectively, this algorithm searches the largest inner,Hadamard, and then outer products at each step. Scales cubically withthe number of terms in the contraction. Equivalent to the ‘optimal’path for most contractions.

Default is ‘greedy’.

Returns:
pathlist of tuples

A list representation of the einsum path.

string_reprstr

A printable representation of the einsum path.

Notes

The resulting path indicates which terms of the input contraction should becontracted first, the result of this contraction is then appended to theend of the contraction list. This list can then be iterated over until allintermediate contractions are complete.

Examples

We can begin with a chain dot example. In this case, it is optimal tocontract theb andc tensors first as represented by the firstelement of the path(1,2). The resulting tensor is added to the endof the contraction and the remaining contraction(0,1) is thencompleted.

>>>np.random.seed(123)>>>a=np.random.rand(2,2)>>>b=np.random.rand(2,5)>>>c=np.random.rand(5,2)>>>path_info=np.einsum_path('ij,jk,kl->il',a,b,c,optimize='greedy')>>>print(path_info[0])['einsum_path', (1, 2), (0, 1)]>>>print(path_info[1])  Complete contraction:  ij,jk,kl->il # may vary         Naive scaling:  4     Optimized scaling:  3      Naive FLOP count:  1.600e+02  Optimized FLOP count:  5.600e+01   Theoretical speedup:  2.857  Largest intermediate:  4.000e+00 elements-------------------------------------------------------------------------scaling                  current                                remaining-------------------------------------------------------------------------   3                   kl,jk->jl                                ij,jl->il   3                   jl,ij->il                                   il->il

A more complex index transformation example.

>>>I=np.random.rand(10,10,10,10)>>>C=np.random.rand(10,10)>>>path_info=np.einsum_path('ea,fb,abcd,gc,hd->efgh',C,C,I,C,C,...optimize='greedy')
>>>print(path_info[0])['einsum_path', (0, 2), (0, 3), (0, 2), (0, 1)]>>>print(path_info[1])  Complete contraction:  ea,fb,abcd,gc,hd->efgh # may vary         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                  current                                remaining--------------------------------------------------------------------------   5               abcd,ea->bcde                      fb,gc,hd,bcde->efgh   5               bcde,fb->cdef                         gc,hd,cdef->efgh   5               cdef,gc->defg                            hd,defg->efgh   5               defg,hd->efgh                               efgh->efgh
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