manifold_dim

t3toolbox.manifold.manifold_dim(s)

def manifold_dim(
        s,  # structure: (shape, tucker_ranks, tt_ranks) = ((N0,...), (n0,...), (1,r1,...,1))
) -> int:  # dimension of the fixed-rank T3 manifold

Get the dimension of the fixed rank T3 manifold with a given structure.

The fixed-rank Tucker tensor train manifold M_{n,r} is described in Appendix A.3 of Alger et al. (2026), “Tucker Tensor Train Taylor Series” (arXiv:2603.21141).

Examples

>>> import numpy as np
>>> import t3toolbox.tucker_tensor_train as t3
>>> import t3toolbox.manifold as t3m
>>> s = ((15,16,13), (9,10,8), (2,7,6,3))
>>> mdim = t3m.manifold_dim(s)
>>> print(mdim)
578

In the following more detailed example, we verify that the manifold dim is correct by generating an excessive number of random dense tangent vectors and performing an SVD on them. The number of nonzero singular values is the dimension of the tangent space, which is the dimension of the manifold.

>>> import numpy as np
>>> import t3toolbox.tucker_tensor_train as t3
>>> import t3toolbox.manifold as t3m
>>> import t3toolbox.frame_variations_format as bvf
>>> s = ((5, 6, 3), (5, 3, 2), (2, 2, 4, 1))
>>> mdim = t3m.manifold_dim(s)
>>> print(mdim)
29
>>> frame, _ = bvf.t3_orthogonal_representations(t3.TuckerTensorTrain.randn(*s))
>>> tucker_shapes, tt_shapes = frame.variation_shapes
>>> n_entries = sum(int(np.prod(sh)) for sh in tucker_shapes) + sum(int(np.prod(sh)) for sh in tt_shapes)
>>> dense_vv = np.stack([t3m.MANIFOLD.randn(frame).to_dense().reshape(-1)
...                      for _ in range(n_entries)])
>>> ss = np.linalg.svd(dense_vv, compute_uv=False)
>>> print(int(np.sum(ss > 1e-9 * ss[0])))   # number of nonzero singular values == manifold_dim
29

Return type:

int