On Tensor-Train Ranks of Tensorized Polynomials

  • Lev VysotskyEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11958)


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.


Tensor train format TT-ranks TT-decomposition Discretization Tensorization 


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Authors and Affiliations

  1. 1.Moscow State UniversityMoscowRussia
  2. 2.Marchuk Institute of Numerical Mathematics of Russian Academy of SciencesMoscowRussia

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