Have a question? Connect with the community at the TensorFlow Forum Visit Forum


Tensor contraction over specified indices and outer product.

Used in the notebooks

Used in the guide Used in the tutorials

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:

$$ C_{i,k} = \sum_j A_{i,j} B_{j,k} $$


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

The corresponding einsum equation is:


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

m0 = tf.random.normal(shape=[2, 3])
m1 = tf.random.normal(shape=[3, 5])
e = tf.einsum('ij,jk->ik', m0, m1)
# output[i,k] = sum_j m0[i,j] * m1[j, k]
(2, 5)

Repeated indices are summed if the output indices are not specified.

e = tf.einsum('ij,jk', m0, m1)  # output[i,k] = sum_j m0[i,j] * m1[j, k]
(2, 5)

Dot product

u = tf.random.normal(shape=[5])
v = tf.random.normal(shape=[5])
e = tf.einsum('i,i->', u, v)  # output = sum_i u[i]*v[i]

Outer product

u = tf.random.normal(shape=[3])
v = tf.random.normal(shape=[5])
e = tf.einsum('i,j->ij', u, v)  # output[i,j] = u[i]*v[j]
(3, 5)


m = tf.ones(2,3)
e = tf.einsum('ij->ji', m0)  # output[j,i] = m0[i,j]
(3, 2)


m = tf.reshape(tf.range(9), [3,3])
diag = tf.einsum('ii->i', m)


# Repeated indices are summed.
trace = tf.einsum('ii', m)  # output[j,i] = trace(m) = sum_i m[i, i]
assert trace == sum(diag)

Batch matrix multiplication

s = tf.random.normal(shape=[7,5,3])
t = tf.random.normal(shape=[7,3,2])
e = tf.einsum('bij,bjk->bik', s, t)
# output[a,i,k] = sum_j s[a,i,j] * t[a, j, k]
(7, 5, 2)

This method does not support broadcasting on named-axes. All axes with matching labels should have the same length. If you have length-1 axes, use tf.squeseze or tf.reshape to eliminate them.

To write code that is agnostic to the number of indices in the input use an ellipsis. The ellipsis is a placeholder for "whatever other indices fit here".

For example, to perform a NumPy-style broadcasting-batch-matrix multiplication where the matrix multiply acts on the last two axes of the input, use:

s = tf.random.normal(shape=[11, 7, 5, 3])
t = tf.random.normal(shape=[11, 7, 3, 2])
e =  tf.einsum('...ij,...jk->...ik', s, t)
(11, 7, 5, 2)

Einsum will broadcast over axes covered by the ellipsis.

s = tf.random.normal(shape=[11, 1, 5, 3])
t = tf.random.normal(shape=[1, 7, 3, 2])
e =  tf.einsum('...ij,...jk->...ik', s, t)
(11, 7, 5, 2)

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.

  • 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.