t3toolbox.tucker_tensor_train.t3_inner_product

t3toolbox.tucker_tensor_train.t3_inner_product(x: t3toolbox.backend.common.typ.Union[TuckerTensorTrain, t3toolbox.backend.common.NDArray], y: t3toolbox.backend.common.typ.Union[TuckerTensorTrain, t3toolbox.backend.common.NDArray], use_orthogonalization: bool = True, use_jax: bool = False)

Compute Hilbert-Schmidt inner product of two Tucker tensor trains.

The Hilbert-Schmidt inner product is defined with respect to the dense N0 x … x N(d-1) tensors that are represented by the Tucker tensor trains, even though these dense tensors are not formed during computations.

For corewise dot product, see t3toolbox.corewise.corewise_dot()

Parameters:

• x (TuckerTensorTrain) – First Tucker tensor train. shape=(N0,…,N(d-1))

• y (TuckerTensorTrain) – Second Tucker tensor train. shape=(N0,…,N(d-1))

• xnp – Linear algebra backend. Default: np (numpy)

Returns:

Hilbert-Schmidt inner product of Tucker tensor trains, (x, y)_HS.

Return type:

scalar

Raises:

ValueError –

• Error raised if either of the TuckerTensorTrains are internally inconsistent

• Error raised if the TuckerTensorTrains have different shapes.

See also

TuckerTensorTrain, t3_shape, t3_add, t3_scale, corewise_dot()

Notes

Algorithm contracts the TuckerTensorTrains in a zippering fashion from left to right.

Examples

>>> import numpy as np
>>> import t3toolbox.tucker_tensor_train as t3
>>> x = t3.t3_corewise_randn((14,15,16), (4,5,6), (1,3,2,1))
>>> y = t3.t3_corewise_randn((14,15,16), (3,7,2), (1,5,6,1))
>>> x_dot_y = t3.t3_inner_product(x, y)
>>> x_dot_y2 = np.sum(x.to_dense() * y.to_dense())
>>> print(np.linalg.norm(x_dot_y - x_dot_y2))
8.731149137020111e-11

Example where leading and trailing TT-ranks are not 1:

>>> import numpy as np
>>> import t3toolbox.tucker_tensor_train as t3
>>> x = t3.t3_corewise_randn((14,15,16), (4,5,6), (2,3,2,2))
>>> y = t3.t3_corewise_randn((14,15,16), (3,7,2), (3,5,6,3))
>>> x_dot_y = t3.t3_inner_product(x, y)
>>> x_dot_y2 = np.sum(x.to_dense() * y.to_dense())
>>> print(np.linalg.norm(x_dot_y - x_dot_y2))
1.3096723705530167e-10

(T3, T3) using stacking:

>>> import numpy as np
>>> import t3toolbox.tucker_tensor_train as t3
>>> x = t3.t3_corewise_randn((14,15,16), (4,5,6), (2,3,2,2), stack_shape=(2,3))
>>> y = t3.t3_corewise_randn((14,15,16), (3,7,2), (3,5,6,3), stack_shape=(2,3))
>>> x_dot_y = t3.t3_inner_product(x, y)
>>> x_dot_y2 = np.sum(x.to_dense() * y.to_dense(), axis=(2,3,4))
>>> print(np.linalg.norm(x_dot_y - x_dot_y2))
2.7761383858792984e-09

Inner product of T3 with dense:

>>> import numpy as np
>>> import t3toolbox.tucker_tensor_train as t3
>>> x = np.random.randn(14,15,16)
>>> y = t3.t3_corewise_randn((14,15,16), (3,7,2), (3,5,6,3))
>>> x_dot_y = t3.t3_inner_product(x, y)
>>> x_dot_y2 = np.sum(x * y.to_dense())
>>> print(np.linalg.norm(x_dot_y - x_dot_y2))
0.0

Inner product of T3 with dense including stacking:

>>> import numpy as np
>>> import t3toolbox.tucker_tensor_train as t3
>>> x = np.random.randn(2,3, 14,15,16)
>>> y = t3.t3_corewise_randn((14,15,16), (3,7,2), (3,5,6,3), stack_shape=(2,3))
>>> x_dot_y = t3.t3_inner_product(x, y)
>>> x_dot_y2 = np.einsum('ijxyz,ijxyz->ij', x, y.to_dense())
>>> print(np.linalg.norm(x_dot_y - x_dot_y2))
1.2014283869232628e-11