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Tensor contraction over specified indices and outer product.
tf.einsum(
equation, *inputs, **kwargs
)
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:
ij,jk->ik
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
.**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').
The contracted Tensor
, with shape determined by equation
.
ValueError
: If
equation
is incorrect,equation
.