Einsum

Tensor contraction according to Einstein summation convention.

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.

Note: This Op is not intended to be called by the user; instead users should call tf.einsum directly. It is a hidden Op used by tf.einsum.

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](http://docs.scipy.org/doc/numpy/user/basics.broadcasting.html).

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.