t3toolbox.manifold.retract
==========================

.. py:function:: 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.

   :param variation: Variation representing the tangent vector we wish to retract to the manifold
   :type variation: T3Variation,
   :param base: Representation of the base point on the manifold where the tangent space is attached.
   :type base: T3Base,
   :param xnp: Linear algebra backend. Default: np (numpy)

   :returns: Retraction of tangent vector onto the manifold.
   :rtype: t3.TuckerTensorTrain

   .. seealso:: :py:obj:`T3Base`, :py:obj:`t3_svd`

   .. rubric:: 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