TuckerTensorTrain.get_minimal_ranks#

static t3toolbox.tucker_tensor_train.TuckerTensorTrain.get_minimal_ranks(shape, tucker_ranks, tt_ranks)#
def get_minimal_ranks(
        shape: Sequence[int],
        tucker_ranks: Sequence[int],
        tt_ranks: Sequence[int],
) -> Tuple[
    Tuple[int, ...],  # new_tucker_ranks
    Tuple[int, ...],  # new_tt_ranks
]:

Find minimal ranks for a hypothetical TuckerTensorTrain with given shape and ranks.

Minimal ranks satisfy:

  • Left TT core unfoldings are full rank: r(i+1) <= (ri*ni)

  • Right TT core unfoldings are full rank: ri <= (ni*r(i+1))

  • Down TT core unfoldings are full rank: ni <= (ri*r(i+1))

  • Tucker ranks do not exceed shape: ni <= Ni

In this function, minimal ranks are defined with respect to a generic Tucker tensor train of the given form based on its structure. We do not account for possible additional rank deficiency due to the numerical values within the cores.

Minimal ranks always exist and are unique.

  • Minimal TT ranks are equal to the ranks of (N*...*Ni) x (N(i+1)*...*N(d-1)) matrix unfoldings.

  • Minimal Tucker ranks are equal to the ranks of Ni x (N1*...*N(i-1)*N(i+1)*...*N(d-1)) matricizations.

Examples

>>> import t3toolbox.tucker_tensor_train as t3
>>> print(t3.TuckerTensorTrain.get_minimal_ranks((10,11,12,13), (14,15,16,17), (98,99,100,101,102)))
((10, 11, 12, 13), (1, 10, 100, 13, 1))
Parameters:

  • shape (collections.abc.Sequence[int])

  • tucker_ranks (collections.abc.Sequence[int])

  • tt_ranks (collections.abc.Sequence[int])

Return type:

Tuple[Tuple[int, Ellipsis], Tuple[int, Ellipsis]]