LSSC 2019: Large-Scale Scientific Computing pp 189-196

# On Tensor-Train Ranks of Tensorized Polynomials

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

## 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.

## Keywords

Tensor train format TT-ranks TT-decomposition Discretization Tensorization

## References

1. 1.
Oseledets, I.V.: Approximation of $$2^d \times 2^d$$ matrices using tensor decomposition. SIAM J. Matrix Anal. Appl. 31(4), 2130–2145 (2010).
2. 2.
Oseledets, I.: Tensor-Train Decomposition. SIAM J. Sci. Comput. 33(5), 2295–2317 (2011).
3. 3.
Grasedyck, L., Hackbusch, W.: An introduction to hierarchical (H-) rank and TT-rank of tensors with examples. Comput. Methods Appl. Math. Comput. Methods Appl. Math. 11(3), 291–304 (2011).
4. 4.
Oseledets, I.: Constructive representation of functions in low-rank tensor formats. Constr. Approx. 37(1), 1–18 (2013).
5. 5.
Tyrtyshnikov, E.E.: Tensor approximations of matrices generated by asymptotically smooth functions. Sb. Math. 194(6), 941–954 (2003).
6. 6.
Khoromskij, B.N.: $$O(d \log N)$$-quantics approximation of $$N$$-$$d$$ tensors in high-dimensional numerical modeling. Constr. Approx. 34(2), 257–280 (2011).
7. 7.
Cormen, T., Leiserson, C., Rivest, R., Stein, C.: Introduction to Algorithms, 3rd edn. The MIT Press, Cambridge (2009)

© Springer Nature Switzerland AG 2020

## Authors and Affiliations

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