# tf.einsum

Tensor contraction over specified indices and outer product.

Einsum allows defining Tensors by defining their element-wise computation. This computation is defined by `equation`, a shorthand form based on Einstein summation. 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, to convert the element-wise equation into the `equation` string, use the following procedure (intermediate strings for matrix multiplication example provided in parentheses):

1. remove variable names, brackets, and commas, (`ik = sum_j ij * jk`)
2. replace "*" with ",", (`ik = sum_j ij , jk`)
3. drop summation signs, and (`ik = ij, jk`)
4. move the output to the right, while replacing "=" with "->". (`ij,jk->ik`)

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 perform batch matrix multiplication with NumPy-style broadcasting across the batch dimensions, use:

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

`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`.
`**kwargs`

• optimize: Optimization strategy to use to find contraction path using opt_einsum. Must be 'greedy', 'optimal', 'branch-2', 'branch-all' or 'auto'. (optional, default: 'greedy').
• name: A name for the operation (optional).

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

`ValueError` If

• the format of `equation` is incorrect,
• number of inputs or their shapes are inconsistent with `equation`.
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