Einsum#

Einsum - 12#

Version

  • name: Einsum (GitHub)

  • domain: main

  • since_version: 12

  • function: False

  • support_level: SupportType.COMMON

  • shape inference: True

This version of the operator has been available since version 12.

Summary

An einsum of the form `term1, term2 -> output-term` produces an output tensor using the following equation

where the reduce-sum performs a summation over all the indices occurring in the input terms (term1, term2)
that do not occur in the output-term.

The Einsum operator evaluates algebraic tensor operations on a sequence of tensors, using the Einstein summation
convention. The equation string contains a comma-separated sequence of lower case letters. Each term corresponds to
an operand tensor, and the characters within the terms correspond to operands dimensions.

This sequence may be followed by "->" to separate the left and right hand side of the equation.
If the equation contains "->" followed by the right-hand side, the explicit (not classical) form of the Einstein
summation is performed, and the right-hand side indices indicate output tensor dimensions. In other cases,
output indices are (implicitly) set to the alphabetically sorted sequence of indices appearing exactly once in the
equation.

When a dimension character is repeated in the left-hand side, it represents summation along the dimension.

The equation may contain ellipsis ("...") to enable broadcasting. Ellipsis must indicate a fixed number of dimensions.
Specifically, every occurrence of ellipsis in the equation must represent the same number of dimensions.
The right-hand side may contain exactly one ellipsis. In implicit mode, the ellipsis dimensions are set to the
beginning of the output. The equation string may contain space (U+0020) character.

Attributes

  • equation (required): Einsum expression string.

Inputs

Between 1 and 2147483647 inputs.

  • Inputs (variadic, heterogeneous) - T: Operands

Outputs

  • Output (heterogeneous) - T: Output tensor

Type Constraints

  • T in ( tensor(double), tensor(float), tensor(float16), tensor(int16), tensor(int32), tensor(int64), tensor(int8), tensor(uint16), tensor(uint32), tensor(uint64), tensor(uint8) ): Constrain input and output types to all numerical tensor types.

Examples

_einsum_transpose

Eqn = 'ij->ji'
node = onnx.helper.make_node(
    'Einsum',
    inputs=['x'],
    outputs=['y'],
    equation=Eqn
)

X = np.random.randn(3, 4)
Y = einsum_reference_implementation(Eqn, (X,))

expect(node, inputs=[X], outputs=[Y], name='test_einsum_transpose')

_einsum_sum

Eqn = 'ij->i'
node = onnx.helper.make_node(
    'Einsum',
    inputs=['x'],
    outputs=['y'],
    equation=Eqn
)

X = np.random.randn(3, 4)
Z = einsum_reference_implementation(Eqn, (X,))

expect(node, inputs=[X], outputs=[Z], name='test_einsum_sum')

_einsum_batch_diagonal

Eqn = '...ii ->...i'
node = onnx.helper.make_node(
    'Einsum',
    inputs=['x'],
    outputs=['y'],
    equation=Eqn
)

X = np.random.randn(3, 5, 5)
Z = einsum_reference_implementation(Eqn, (X,))

expect(node, inputs=[X], outputs=[Z], name='test_einsum_batch_diagonal')

_einsum_inner_prod

Eqn = 'i,i'
node = onnx.helper.make_node(
    'Einsum',
    inputs=['x', 'y'],
    outputs=['z'],
    equation=Eqn
)

X = np.random.randn(5)
Y = np.random.randn(5)
Z = einsum_reference_implementation(Eqn, (X, Y))

expect(node, inputs=[X, Y], outputs=[Z], name='test_einsum_inner_prod')

_einsum_batch_matmul

Eqn = 'bij, bjk -> bik'
node = onnx.helper.make_node(
    'Einsum',
    inputs=['x', 'y'],
    outputs=['z'],
    equation=Eqn
)

X = np.random.randn(5, 2, 3)
Y = np.random.randn(5, 3, 4)
Z = einsum_reference_implementation(Eqn, (X, Y))

expect(node, inputs=[X, Y], outputs=[Z], name='test_einsum_batch_matmul')