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Explicit and structured sum of squares lower bounds from high-dimensional expanders

Irit Dinur, Yuval Filmus, Prahladh Harsha and Madhur Tulsiani
ITCS 2021

We construct an explicit family of 3XOR instances which are hard for $O(\sqrt{\log n})$ levels of Sum-of-Squares. Our constructions are based on the LSV complexes, and rely on two of their crucial properties: cosystolic expansion and local nonpositive curvature (via Gromov’s filling inequality, which generalizes the isoperimetric inequality in $\mathbb{R}^n$)).

In contrast to many other constructions, our variables correspond to edges in the complex. Curiously, Alev, Jeronimo and Tulsiani showed that if variables correspond to vertices, then instances based on high-dimensional expanders are easy.

BibTeX

@inproceedings{DFHT2021,
 title = {Explicit and structured sum of squares lower bounds from high-dimensional expanders},
 author = {Irit Dinur and Yuval Filmus and Prahladh Harsha and Madhur Tulsiani},
 booktitle = {12th Innovations in Theoretical Computer Science Conference (ITCS 2021)},
 year = {2021}
}
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