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DOI: 10.4230/LIPIcs.APPROX-RANDOM.2019.40

Direct Sum Testing: The General Case

Authors: Irit Dinur and Konstantin Golubev

Published in: LIPIcs, Volume 145, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019)

Abstract

A function f:[n_1] x ... x[n_d]->F_2 is a direct sum if it is of the form f (a_1,...,a_d) = f_1(a_1) oplus ... oplus f_d (a_d), for some d functions f_i:[n_i]->F_2 for all i=1,..., d, and where n_1,...,n_d in N. We present a 4-query test which distinguishes between direct sums and functions that are far from them. The test relies on the BLR linearity test (Blum, Luby, Rubinfeld, 1993) and on the direct product test constructed by Dinur & Steurer (2014). We also present a different test, which queries the function (d+1) times, but is easier to analyze. In multiplicative +/- 1 notation, this reads as follows. A d-dimensional tensor with +/- 1 entries is called a tensor product if it is a tensor product of d vectors with +/- 1 entries, or equivalently, if it is of rank 1. The presented tests can be read as tests for distinguishing between tensor products and tensors that are far from being tensor products.

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Irit Dinur and Konstantin Golubev. Direct Sum Testing: The General Case. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 145, pp. 40:1-40:11, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{dinur_et_al:LIPIcs.APPROX-RANDOM.2019.40,
  author =	{Dinur, Irit and Golubev, Konstantin},
  title =	{{Direct Sum Testing: The General Case}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019)},
  pages =	{40:1--40:11},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-125-2},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{145},
  editor =	{Achlioptas, Dimitris and V\'{e}gh, L\'{a}szl\'{o} A.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2019.40},
  URN =		{urn:nbn:de:0030-drops-112554},
  doi =		{10.4230/LIPIcs.APPROX-RANDOM.2019.40},
  annote =	{Keywords: property testing, direct sum, tensor product}
}

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Direct Sum Testing: The General Case

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