t3toolbox.manifold.retract#

t3toolbox.manifold.retract(variation: t3toolbox.basis_coordinates_format.T3Variation, base: t3toolbox.basis_coordinates_format.T3Base, use_jax: bool = False) → t3toolbox.tucker_tensor_train.TuckerTensorTrain#

Retract Tucker tensor train tangent vector to manifold.

Parameters:

  • variation (T3Variation,) – Variation representing the tangent vector we wish to retract to the manifold

  • base (T3Base,) – Representation of the base point on the manifold where the tangent space is attached.

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

Returns:

Retraction of tangent vector onto the manifold.

Return type:

t3.TuckerTensorTrain

See also

T3Base, t3_svd

Examples

>>> import numpy as np
>>> import t3toolbox.tucker_tensor_train as t3
>>> import t3toolbox.manifold as t3m
>>> import t3toolbox.common
>>> import t3toolbox.orthogonalization as orth
>>> import t3toolbox.corewise as cw
>>> p = t3.t3_corewise_randn(((14,15,16), (4,5,6), (1,3,2,1)))
>>> base, _ = orth.orthogonal_representations(p)
>>> variation = t3m.tangent_randn(base) # Random tangent vector
>>> ret_v = t3m.retract(variation, base) # Retract tangent vector to manifold
>>> ret_V = t3.t3_to_dense(ret_v)
>>> V = t3m.tangent_to_dense(variation, base, include_shift=True)
>>> print(np.linalg.norm(ret_V - V)) # vector changes
0.14335564543255402
>>> v2 = cw.corewise_scale(variation, 1e-2) # make the tangent vector shorter for smaller retraction
>>> ret_v2 = t3m.retract(v2, base)
>>> ret_V2 = t3.t3_to_dense(ret_v2)
>>> V2 = t3m.tangent_to_dense(v2, base, include_shift=True)
>>> print(np.linalg.norm(ret_V2 - V2)) # vector changes
4.9488133126395654e-05