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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:


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 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.

  • 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.