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# tf.einsum

Tensor contraction over specified indices and outer product.

This function returns a tensor whose elements are defined by `equation`, which is written in a shorthand form inspired by the Einstein summation convention. As an example, consider multiplying two matrices A and B to form a matrix C. The elements of C are given by:

``````  C[i,k] = sum_j A[i,j] * B[j,k]
``````

The corresponding `equation` is:

``````  ij,jk->ik
``````

In general, the `equation` is obtained from the more familiar element-wise equation by

1. removing variable names, brackets, and commas,
2. replacing "*" with ",",
3. dropping summation signs, and
4. moving the output to the right, and replacing "=" with "->".

Many common operations can be expressed in this way. For example:

``````# Matrix multiplication
>>> einsum('ij,jk->ik', m0, m1)  # output[i,k] = sum_j m0[i,j] * m1[j, k]

# Dot product
>>> einsum('i,i->', u, v)  # output = sum_i u[i]*v[i]

# Outer product
>>> einsum('i,j->ij', u, v)  # output[i,j] = u[i]*v[j]

# Transpose
>>> einsum('ij->ji', m)  # output[j,i] = m[i,j]

# Trace
>>> einsum('ii', m)  # output[j,i] = trace(m) = sum_i m[i, i]

# Batch matrix multiplication
>>> einsum('aij,ajk->aik', s, t)  # out[a,i,k] = sum_j s[a,i,j] * t[a, j, k]
``````

To enable and control broadcasting, use an ellipsis. For example, to do batch matrix multiplication, you could use:

````einsum('...ij,...jk->...ik', u, v)`
```

This function behaves like `numpy.einsum`, but does not support:

• Subscripts where an axis appears more than once for a single input (e.g. `ijj,k->ik`) unless it is a trace (e.g. `ijji`).

`equation` a `str` describing the contraction, in the same format as `numpy.einsum`.
`*inputs` the inputs to contract (each one a `Tensor`), whose shapes should be consistent with `equation`.
`name` A name for the operation (optional).

The contracted `Tensor`, with shape determined by `equation`.

`ValueError` If

• the format of `equation` is incorrect,
• the number of inputs implied by `equation` does not match `len(inputs)`,
• an axis appears in the output subscripts but not in any of the inputs,
• the number of dimensions of an input differs from the number of indices in its subscript, or
• the input shapes are inconsistent along a particular axis.
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