TuckerTensorTrain.norm

t3toolbox.tucker_tensor_train.TuckerTensorTrain.norm(use_orthogonalization=True)

def norm(
        self,
        use_orthogonalization: bool = True, # for numerical stability
):

Compute Hilbert-Schmidt (Frobenius) norm of this TuckerTensorTrain.

The Hilbert-Schmidt norm is defined with respect to the dense N0 x ... x N(d-1) tensor that is represented by the TuckerTensorTrain.

x.norm() = np.linalg.norm(x.to_dense())

For corewise norm, see t3toolbox.corewise.corewise_norm()

Parameters:

use_orthogonalization (bool, optional) – If True, compute norm by orthogonalizing (more stable). If False, compute norm with conventional zippering (faster, more suited for automatic differentiation). Default: use_orthogonalization=True.

Returns:

result – Hilbert-Schmidt (Frobenius) norm of Tucker tensor train, ||x||_HS. If stacked, result.shape=self.stack_shape. Otherwise, result is scalar.

Return type:

scalar or NDArray

See also

TuckerTensorTrain.__add__(), TuckerTensorTrain.__sub__(), TuckerTensorTrain.__neg__(), TuckerTensorTrain.__mul__(), TuckerTensorTrain.inner(), TuckerTensorTrain.sum()

Examples

>>> import numpy as np
>>> import t3toolbox.tucker_tensor_train as t3
>>> np.random.seed(0)
>>> x = t3.TuckerTensorTrain.randn((14,15,16), (4,5,6), (2,3,2,2))
>>> print(np.allclose(x.norm(), np.linalg.norm(x.to_dense())))
True

Stacked – norm() returns an array of shape stack_shape, one norm per slice:

>>> import numpy as np
>>> import t3toolbox.tucker_tensor_train as t3
>>> np.random.seed(0)
>>> x = t3.TuckerTensorTrain.randn((14,15,16), (4,5,6), (2,3,2,2), stack_shape=(2,3))
>>> norms_x = x.norm(use_orthogonalization=True)
>>> print(norms_x.shape)
(2, 3)
>>> x_dense = x.to_dense()
>>> norms_x_dense = np.sqrt(np.sum(x_dense**2, axis=(-3,-2,-1)))
>>> print(np.allclose(norms_x, norms_x_dense))
True