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On the sum-of-products to product-of-sums transformation between analytical low-rank approximations in finite basis representation

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Abstract

In this work, we analyze and compare different possible strategies for the transformations among low-rank (i.e., few number of terms) tensor approximations. The motivation behind this is to achieve compact yet accurate representations of potential-like operators (scalar fields) in symbolic or analytical form. We do this analysis from a formal and from a numerical perspective. Specifically, we concentrate on Tucker and Canonic Polyadic ansätze. We introduce the sum-of-product finite basis representations (SOP-FBR) for both. Here, the factor matrices (aka single-particle functions) are approximated through a set of auxiliary basis functions, specific to the system. In this way, analytical, grid-independent, low-rank expressions can be obtained. We illustrate how finite-precision arithmetic hinders transformations among all these forms. The solution to this issue seems to adapt current algorithms to high-precision arithmetic at the expense of an increase in CPU times.

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Data availability

The data that support the findings of this study are available from the corresponding author upon reasonable request.

Notes

  1. (see https://numpy.org/doc/stable/user/basics.types.html#extended-precision).

  2. (https://docs.python.org/3/library/math.html).

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Acknowledgements

NN is very grateful for its Ph.D. funding from the University Paris-Saclay. The authors are very thankful to J.-Y. Bazzara and A. Borissov (ISMO) for their computer support.

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Authors and Affiliations

  1. Faculty of Chemistry and Food Chemistry, Technische Universität Dresden, 01069, Dresden, Germany

    Ramón L. Panadés-Barrueta

  2. CNRS, Institut des Sciences Moléculaires d’Orsay, Université Paris-Saclay, 91405, Orsay, France

    Natasa Nadoveza, Fabien Gatti & Daniel Peláez

Authors
  1. Ramón L. Panadés-Barrueta
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  2. Natasa Nadoveza
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  3. Fabien Gatti
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  4. Daniel Peláez
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Correspondence to Ramón L. Panadés-Barrueta or Daniel Peláez.

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Panadés-Barrueta, R.L., Nadoveza, N., Gatti, F. et al. On the sum-of-products to product-of-sums transformation between analytical low-rank approximations in finite basis representation. Eur. Phys. J. Spec. Top. 232, 1897–1904 (2023). https://doi.org/10.1140/epjs/s11734-023-00928-z

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