t3toolbox.tucker_tensor_train.t3svd#

t3toolbox.tucker_tensor_train.t3svd(x: TuckerTensorTrain, min_tt_ranks: t3toolbox.backend.common.typ.Sequence[int] = None, min_tucker_ranks: t3toolbox.backend.common.typ.Sequence[int] = None, max_tt_ranks: t3toolbox.backend.common.typ.Sequence[int] = None, max_tucker_ranks: t3toolbox.backend.common.typ.Sequence[int] = None, rtol: float = None, atol: float = None, squash_tails_first: bool = True, use_jax: bool = False) → t3toolbox.backend.common.typ.Tuple[TuckerTensorTrain, t3toolbox.backend.common.typ.Tuple[t3toolbox.backend.common.NDArray, Ellipsis], t3toolbox.backend.common.typ.Tuple[t3toolbox.backend.common.NDArray, Ellipsis]]#

Compute (truncated) T3-SVD of TuckerTensorTrain.

Parameters:

  • x (TuckerTensorTrain) – The Tucker tensor train. structure=((N1,…,Nd), (n1,…,nd), (1,r1,…r(d-1),1))

  • min_tucker_ranks (typ.Sequence[int]) – Minimum Tucker ranks for truncation.

  • min_tt_ranks (typ.Sequence[int]) – Minimum TT-ranks for truncation.

  • max_tucker_ranks (typ.Sequence[int]) – Maximum Tucker ranks for truncation.

  • max_tt_ranks (typ.Sequence[int]) – Maximum TT-ranks for truncation.

  • rtol (float) – Relative tolerance for truncation.

  • atol (float) – Absolute tolerance for truncation.

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

Returns:

  • NDArray – New TuckerTensorTrain representing the same tensor (or a truncated version), but with modified cores

  • typ.Tuple[NDArray,…] – Singular values associated with edges between Tucker cores and TT-cores

  • typ.Tuple[NDArray,…] – Singular values associated with edges between adjacent TT-cores

See also

left_svd_3tensor, right_svd_3tensor, outer_svd_3tensor, up_svd_ith_tucker_core, left_svd_ith_tt_core, right_svd_ith_tt_core, up_svd_ith_tt_core, down_svd_ith_tt_core, truncated_svd

Examples

T3-SVD with no truncation: (ranks may decrease to minimal values, but no approximation error)

>>> import numpy as np
>>> import t3toolbox.tucker_tensor_train as t3
>>> x = t3.t3_corewise_randn((5,6,3), (4,4,3), (1,3,2,1))
>>> x2, ss_tucker, ss_tt = t3.t3svd(x) # Compute T3-SVD
>>> x_dense = x.to_dense()
>>> x2_dense = x2.to_dense()
>>> print(np.linalg.norm(x_dense - x2_dense)) # Tensor unchanged
7.556835759880194e-13
>>> ss_tt1 = np.linalg.svd(x_dense.reshape((5, 6*3)))[1] # Singular values of unfolding 1
>>> print(ss_tt1); print(ss_tt[1])
[1.75326490e+02 3.41363029e+01 9.31164204e+00 1.33610061e-14 4.11601708e-15]
[175.32648969  34.13630287   9.31164204]
>>> ss_tucker2 = np.linalg.svd(x_dense.transpose([2,0,1]).reshape((3,5*6)))[1] # Singular values of matricization 2
>>> print(ss_tucker2); print(ss_tucker[2])
[1.71350937e+02 5.12857505e+01 1.36927051e-14]
[171.35093708  51.28575045]

T3-SVD with truncation based on relative tolerance:

>>> import numpy as np
>>> import t3toolbox.tucker_tensor_train as t3
>>> B0 = np.random.randn(35,40) @ np.diag(1.0 / np.arange(1, 41)**2) # preconditioned indices
>>> B1 = np.random.randn(45,50) @ np.diag(1.0 / np.arange(1, 51)**2)
>>> B2 = np.random.randn(55,60) @ np.diag(1.0 / np.arange(1, 61)**2)
>>> G0 = np.random.randn(1,35,30)
>>> G1 = np.random.randn(30,45,40)
>>> G2 = np.random.randn(40,55,1)
>>> tucker_cores_x = (B0, B1, B2)
>>> tt_cores_x = (G0, G1, G2)
>>> x = t3.TuckerTensorTrain(tucker_cores_x, tt_cores_x) # Tensor has spectral decay due to preconditioning
>>> x2, ss_tucker, ss_tt = t3.t3svd(x, rtol=1e-2) # Truncate singular values to reduce rank
>>> print(x.uniform_structure)
((40, 50, 60), (35, 45, 55), (1, 30, 40, 1), ())
>>> print(x2.uniform_structure)
((40, 50, 60), (6, 6, 5), (1, 6, 5, 1), ())
>>> x_dense = x.to_dense()
>>> x2_dense = x2.to_dense()
>>> print(np.linalg.norm(x_dense - x2_dense)/np.linalg.norm(x_dense)) # Should be near rtol=1e-2
0.01919726997372989

T3-SVD with truncation based on absolute tolerance:

>>> import numpy as np
>>> import t3toolbox.tucker_tensor_train as t3
>>> x = t3.t3_corewise_randn((14,15,16), (10,11,12), (1,8,9,1))
>>> x2, ss_tucker, ss_tt = t3.t3svd(x, max_tucker_ranks=(3,3,3), max_tt_ranks=(1,2,2,1)) # Truncate based on ranks
>>> print(x.uniform_structure)
((14, 15, 16), (10, 11, 12), (1, 8, 9, 1), ())
>>> print(x2.uniform_structure)
((14, 15, 16), (3, 3, 2), (1, 2, 2, 1), ())

Example where first and last ranks are not ones:

>>> import numpy as np
>>> import t3toolbox.tucker_tensor_train as t3
>>> x = t3.t3_corewise_randn((5,6,3), (4,4,3), (2,3,2,2))
>>> x2, ss_tucker, ss_tt = t3.t3svd(x, squash_tails_first=False) # Compute T3-SVD
>>> x_dense = x.to_dense(squash_tails=False)
>>> x2_dense = x2.to_dense(squash_tails=False)
>>> print(np.linalg.norm(x_dense - x2_dense)) # Tensor unchanged
5.486408687260824e-13
>>> ss_tt0 = np.linalg.svd(x_dense.reshape((2,5*6*3*2)))[1] # Singular values of leading unfolding
>>> print(ss_tt0); print(ss_tt[0])
[303.0474449   88.85034392]
[303.0474449   88.85034392]
>>> ss_tt3 = np.linalg.svd(x_dense.reshape((2*5*6*3,2)))[1] # Singular values of trailing unfolding
>>> print(ss_tt3); print(ss_tt[3])
[299.45433768 100.29574828]
[299.45433768 100.29574828]