Approximation and composition of functions in quantized tensor trains via orthogonal polynomial expansions
- 1Institute of Fundamental Physics IFF-CSIC, Calle Serrano 113b, Madrid, 28006, Madrid, Spain
Abstract
This work explores the representation of univariate and multivariate functions as quantized tensor trains (QTT). It develops a constructive algorithm that employs expansions in orthogonal polynomial bases and Clenshaw evaluations to represent analytic and highly differentiable functions as QTT approximants. This approach provides a general framework for function composition in QTT form that generalizes efficiently to multidimensional settings. Without loss of generality, the presentation focuses on the Chebyshev basis, where the method demonstrates rapid convergence for highly differentiable functions, in agreement with classical theoretical predictions, and remains numerically stable at high polynomial orders. The performance of the algorithm is compared with that of tensor cross-interpolation (TCI) and multiscale interpolative constructions, demonstrating competitive performance and favorable scaling behavior in certain scenarios.
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BibTeX
@article{rodriguezaldavero2026approximation,
title = {{Approximation and composition of functions in quantized tensor trains via orthogonal polynomial expansions}},
author = {Juan José Rodríguez-Aldavero and Paula García-Molina and Luca Tagliacozzo and Juan José García‐Ripoll},
journal = {Linear Algebra and its Applications},
year = {2026},
month = aug,
doi = {10.1016/j.laa.2026.08.022},
url = {https://doi.org/10.1016/j.laa.2026.08.022},
note = {CC BY},
}
RIS
TY - JOUR
TI - Approximation and composition of functions in quantized tensor trains via orthogonal polynomial expansions
AU - Juan José Rodríguez-Aldavero
AU - Paula García-Molina
AU - Luca Tagliacozzo
AU - Juan José García‐Ripoll
PY - 2026
DA - 2026/08/19
JO - Linear Algebra and its Applications
DO - 10.1016/j.laa.2026.08.022
UR - https://doi.org/10.1016/j.laa.2026.08.022
AB - This work explores the representation of univariate and multivariate functions as quantized tensor trains (QTT). It develops a constructive algorithm that employs expansions in orthogonal polynomial bases and Clenshaw evaluations to represent analytic and highly differentiable functions as QTT approximants. This approach provides a general framework for function composition in QTT form that generalizes efficiently to multidimensional settings. Without loss of generality, the presentation focuses on the Chebyshev basis, where the method demonstrates rapid convergence for highly differentiable functions, in agreement with classical theoretical predictions, and remains numerically stable at high polynomial orders. The performance of the algorithm is compared with that of tensor cross-interpolation (TCI) and multiscale interpolative constructions, demonstrating competitive performance and favorable scaling behavior in certain scenarios.
SN - 0024-3795
SN - 1873-1856
ER -