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On Tensor-Train Ranks of Tensorized Polynomials
Discretization followed by tensorization (mapping from low-
dimensional to high-dimensional data) can be used to construct low-
parametric approximations of functions. For example, a function f
defined on [0, 1] may be mapped to a d-dimensional tensor A ∈ R b×···×b
with elements A(i 1 , . . . , i d ) = f (i_1 b^{−1} + · · · + i_d b^{−d} ), i k ∈ {0, . . . , b − 1}.
The tensor A can now be compressed using one of the tensor formats,
e.g. tensor train format. It has been noticed in practice that approximate
TT-ranks of tensorizations of degree-n polynomials grow very slowly with
respect to n, while the only known bound for them is n + 1. In this paper
we try to explain the observed effect. New bounds of the described TT-
ranks are proved and shown experimentally to quite successfully capture
the observed distribution of ranks.