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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)