TY - GEN
T1 - Direct sum testing
T2 - 22nd International Conference on Approximation Algorithms for Combinatorial Optimization Problems and 23rd International Conference on Randomization and Computation, APPROX/RANDOM 2019
AU - Dinur, Irit
AU - Golubev, Konstantin
N1 - Publisher Copyright: © Irit Dinur and Konstantin Golubev.
PY - 2019/9
Y1 - 2019/9
N2 - A function f : [n1] × · · · × [nd] → F2 is a direct sum if it is of the form f (a1,..., ad) = f1(a1) ⊕ ... ⊕ fd(ad), for some d functions fi : [ni] → F2 for all i = 1, ..., d, and where n1, ..., nd ∊ 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.
AB - A function f : [n1] × · · · × [nd] → F2 is a direct sum if it is of the form f (a1,..., ad) = f1(a1) ⊕ ... ⊕ fd(ad), for some d functions fi : [ni] → F2 for all i = 1, ..., d, and where n1, ..., nd ∊ 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.
UR - http://www.scopus.com/inward/record.url?scp=85072870692&partnerID=8YFLogxK
U2 - https://doi.org/10.4230/LIPIcs.APPROX-RANDOM.2019.40
DO - https://doi.org/10.4230/LIPIcs.APPROX-RANDOM.2019.40
M3 - منشور من مؤتمر
T3 - Leibniz International Proceedings in Informatics, LIPIcs
BT - Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, APPROX/RANDOM 2019
A2 - Achlioptas, Dimitris
A2 - Vegh, Laszlo A.
PB - Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
Y2 - 20 September 2019 through 22 September 2019
ER -