On Tensor-Train Ranks of Tensorized Polynomials

Abstract

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 $A \in \mathbb {R}^{b\times \dots \times b}$ with elements $A(i_1,\dots ,i_d) = f(i_1b^{-1} + \dots + i_db^{-d})$, $i_k \in \{0,\dots ,b-1\}$. 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.

The work was supported by the Russian Science Foundation, grant 19-11-00338.