Tensor contraction according to Einstein summation convention.
tf.raw_ops.Einsum(
inputs, equation, name=None
)
Implements generalized Tensor contraction and reduction. Each input Tensor must
have a corresponding input subscript appearing in the comma-separated left-hand
side of the equation. The right-hand side of the equation consists of the
output subscript. The input subscripts and the output subscript should consist
of zero or more named axis labels and at most one ellipsis (...).
The named axis labels may be any single character other than those having
special meaning, namely ,.->. The behavior of this Op is undefined if it
receives an ill-formatted equation; since the validation is done at
graph-building time, we omit format validation checks at runtime.
Operations are applied to the input(s) according to the following rules:
(a) Generalized Diagonals: For input dimensions corresponding to axis labels
appearing more than once in the same input subscript, we take the
generalized (k-dimensional) diagonal.
For example, in the equation iii->i with input shape [3, 3, 3], the
generalized diagonal would consist of 3 elements at indices (0, 0, 0),
(1, 1, 1) and (2, 2, 2) to create a Tensor of shape [3].
(b) Reduction: Axes corresponding to labels appearing only in one input
subscript but not in the output subscript are summed over prior to Tensor
contraction.
For example, in the equation ab,bc->b, the axis labels a and c are
the reduction axis labels.
(c) Batch Dimensions: Axes corresponding to labels appearing in each of the
input subscripts and also in the output subscript make up the batch
dimensions in Tensor contraction. Unnamed axis labels corresponding to
ellipsis (...) also correspond to batch dimensions.
For example, for the equation denoting batch matrix multiplication,
bij,bjk->bik, the axis label b corresponds to a batch dimension.
(d) Contraction: In case of binary einsum, axes corresponding to labels
appearing in two different inputs (and not in the output) are contracted
against each other.
Considering the batch matrix multiplication equation again
(bij,bjk->bik), the contracted axis label is j.
(e) Expand Diagonal: If the output subscripts contain repeated (explicit) axis
labels, the opposite operation of (a) is applied. For example, in the
equation i->iii, and input shape [3], the output of shape [3, 3, 3]
are all zeros, except for the (generalized) diagonal which is populated
with values from the input.
Note: This operation is not supported by np.einsum or tf.einsum; it is
provided to enable computing the symbolic gradient of tf.einsum.
The output subscripts must contain only labels appearing in at least one of the input subscripts. Furthermore, all dimensions mapping to the same axis label must be equal.
Any of the input and output subscripts may contain at most a single ellipsis
(...). These ellipsis are mapped against dimensions not corresponding to any
named axis label. If two inputs contain ellipsis, then they are broadcasted
according to standard NumPy broadcasting
rules.
The broadcasted dimensions are placed in the corresponding location of the ellipsis in the output subscript. If the broadcasted dimensions are non-empty and the output subscripts do not contain ellipsis, then an InvalidArgument error is raised.
Returns | |
|---|---|
A Tensor. Has the same type as inputs.
|
numpy compatibility
Similar to numpy.einsum.
Comparison with numpy.einsum:
- This Op only supports unary and binary forms of
numpy.einsum. - This Op does not support implicit form. (i.e. equations without
->). This Op also supports repeated indices in the output subscript, which is not supported by
numpy.einsum.