t3toolbox.manifold.project_t3_onto_tangent_space#

t3toolbox.manifold.project_t3_onto_tangent_space(x: t3toolbox.tucker_tensor_train.TuckerTensorTrain | t3toolbox.OLD_uniform.UniformTuckerTensorTrain, orthogonal_base: t3toolbox.basis_coordinates_format.T3Base | t3toolbox.OLD_uniform.UniformT3Base, use_jax: bool = False) → t3toolbox.basis_coordinates_format.T3Variation#

Projects TuckerTensorTrain onto tangent space to the manifold of fixed rank TuckerTensorTrains.

Parameters:

  • x (t3.TuckerTensorTrain) – TuckerTensorTrain to project

  • orthogonal_base (T3Base) – Minimal rank orthogonal representations of base point on manifold where tangent space is attached

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

Returns:

Gauged variation representing the orthogonal projection of x onto the tangent space.

Return type:

T3Variation

See also

T3Base, oblique_gauge_projection, orthogonal_gauge_projection, ADD

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), (5,3,2,4)))
>>> base, _ = orth.orthogonal_representations(p)
>>> x = t3.t3_corewise_randn(((14,15,16), (7,4,8), (3,5,4,2)))
>>> proj_x = t3m.project_t3_onto_tangent_space(x, base) # Project x onto tangent space
>>> P = t3.t3_to_dense(p)
>>> X = t3.t3_to_dense(x)
>>> proj_X = t3m.tangent_to_dense(proj_x, base)
>>> print(np.sum((X - proj_X) * (proj_X - P)) / np.sum(X)) # Check that x was projected orthogonally
-2.7295025395842007e-13

Uniform example: DOESNT WORK YET

>>> import numpy as np
>>> import t3toolbox.tucker_tensor_train as t3
>>> import t3toolbox.manifold as t3m
>>> import t3toolbox.uniform as ut3
>>> import t3toolbox.uniform_manifold as utm
>>> import t3toolbox.orthogonalization as orth
>>> import t3toolbox.t3svd as t3svd
>>> p = t3.t3_corewise_randn(((14,15,16), (4,5,6), (5,3,2,4)))
# >>> p = t3.t3_corewise_randn(((15,15,15), (5,5,5), (1,3,3,1)))
>>> p, _, _ = t3svd.t3svd(p)
>>> base, dummy_var = orth.orthogonal_representations(p)
>>> x = t3.t3_corewise_randn(((14,15,16), (7,4,8), (3,5,4,2)))
# >>> x = t3.t3_corewise_randn(((15,15,15), (5,5,5), (1,3,3,1)))
>>> x, _, _ = t3svd.t3svd(x)
>>> proj_x = t3m.project_t3_onto_tangent_space(x, base) # Project x onto tangent space
>>> dense_proj_x = t3m.tangent_to_dense(proj_x, base)
>>> _, uniform_base, bv_mask = ut3.bv_to_ubv(dummy_var, base)
>>> uniform_x, x_mask = ut3.t3_to_ut3(x)
>>> uniform_proj_x = t3m.project_t3_onto_tangent_space(uniform_x, uniform_base) # Project x onto tangent space
>>> dense_uniform_proj_x = utm.uniform_tangent_to_dense(uniform_proj_x, uniform_base, bv_mask)
>>> print(np.linalg.norm(dense_uniform_proj_x - dense_proj_x))