t3toolbox.manifold.tangent_to_t3#

t3toolbox.manifold.tangent_to_t3(variation: t3toolbox.basis_coordinates_format.T3Variation, base: t3toolbox.basis_coordinates_format.T3Base, include_shift: bool = False, use_jax: bool = False) → t3toolbox.tucker_tensor_train.TuckerTensorTrain#

Rank 2r Tucker tensor train representation of tangent vector.

Without shift, we use the formula:

v(x,y,z,w) = ([dU1(B x) L1(B x)]) ([R2(B y)        0]) ([R3(B z)        0]) ([R4(B w) ])
             (                  ) ([dU2(B y) L2(B y)]) ([dU3(B z) L3(B z)]) ([dU4(B w)])
             (         +        ) (         +        ) (        +         ) (    +     )
             ([O1(dB x)       0]) ([0              0]) ([0              0]) ([0       ])
             (                  ) ([O2(dB y)       0]) ([O3(dB z)       0]) ([O4(dB w)])

With shift is same as unshifted, except last backend modified as follows:

[R4(B w) ]                  [R4(B w)           ]
[dU4(B w)]                  [L4(B w) + dU4(B w)]
    +             ->            +
[0       ]                  [0                 ]
[O4(dB w)]                  [O4(dB w)          ]
Parameters:

  • variation (T3Variation,) – Variation representing the tangent vector

  • base (T3Base,) – Representation of the base point at which the tangent space attaches to the manifold.

  • include_shift (bool) – If False, return tangent vector v only. If True, shift tangent vector so it is attached at the base point, p+v.

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

Returns:

Tucker tensor train representation of tangent vector, which has doubled ranks

Return type:

TuckerTensorTrain

See also

T3Base, T3Variation, TuckerTensorTrain

Examples

>>> import numpy as np
>>> import t3toolbox.tucker_tensor_train as t3
>>> import t3toolbox.manifold as t3m
>>> import t3toolbox.orthogonalization as orth
>>> p = t3.t3_corewise_randn(((14,15,16), (4,5,6), (2,3,2,2)))
>>> base, _ = orth.orthogonal_representations(p)
>>> variation = t3m.tangent_randn(base)
>>> v_t3 = t3m.tangent_to_t3(variation, base) # tangent vector only (attached at zero)
>>> v_dense = t3.t3_to_dense(v_t3)
>>> v_dense2 = t3m.tangent_to_dense(variation, base)
>>> print(np.linalg.norm(v_dense - v_dense2))
2.678565538404836e-15
>>> p_plus_v_t3 = t3m.tangent_to_t3(variation, base, include_shift=True) # shifted tangent vector (include attachment at base point)
>>> p_plus_v_dense = t3.t3_to_dense(p_plus_v_t3)
>>> p_plus_v_dense2 = v_dense2 + t3.t3_to_dense(p)
>>> print(np.linalg.norm(p_plus_v_dense - p_plus_v_dense2))
1.2102169224182523e-12