Monthly Archives: October 2018

News for September 2018

Last month was slower than usual, with only one property testing paper*

Property testing and expansion in cubical complexes, by David Garber, Uzi Vishne (arXiv). Consider the question of testing if an arbitrary function \(f\colon V\times V \to\{-1,1\}\) is of the form \(f(x,y) = h(x)h(y)\) for some \(h\colon V\to\{-1,1\}\). An intuitive one-sided test, shown to work by Lubotzky and Kaufman (2014), is to pick uniformly random \(x,y,z\in V\) and check that \(f(x,y)f(y,z)f(z,x)=1\). This paper considers the high-dimensional generalization of testing the property that a function\(f\colon V\times V \times V\times V \to\{-1,1\}\) is of the form \(f(w,x,y,z) = \alpha\cdot h(w,x)h(y,x)h(y,z) h(w,z)\), for some \(h\colon V\times V\to\{-1,1\}\) and sign \(\alpha\in\{-1,1\}\). The authors derive necessary and sufficient conditions for testability of this property, by formulating it in the language of incidence geometry and exploiting this connection.

* If we missed any, please let us know in the comments.