tf.einsum

TensorFlow 1 version

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Tensor contraction over specified indices and outer product.

tf.einsum(
    equation, *inputs, **kwargs
)

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)