Communication: Hypercontractive inequality for pseudo-Boolean functions of bounded Fourier width

• Authors: Gregory Gutin;Anders Yeo
• Venue: Discrete Applied Mathematics
• Year: 2012

A function f:{-1,1}^n-R is called pseudo-Boolean. It is well-known that each pseudo-Boolean function f can be written as f(x)=@?"I"@?"Ff@?(I)@g"I(x), where F@?{I:I@?[n]}, [n]={1,2,...,n}, @g"I(x)=@?"i"@?"Ix"i and f@?(I) are non-zero reals. The degree of f is max{|I|:I@?F} and the width of f is the minimum integer @r such that every i@?[n] appears in at most @r sets in F. For i@?[n], let x"i be a random variable taking values 1 or -1 uniformly and independently from all other variables x"j, ji. Let x=(x"1,...,x"n). The p-norm of f is @?f@?"p=(E[|f(x)|^p])^1^/^p for any p=1. It is well-known that @?f@?"q=@?f@?"p whenever qp=1. However, the higher norm can be bounded by the lower norm times a coefficient not directly depending on f: if f is of degree d and qp1 then @?f@?"q@?(q-1p-1)^d^/^2@?f@?"p. This inequality is called the Hypercontractive Inequality. We show that one can replace d by @r in the Hypercontractive Inequality for each qp=2 as follows: @?f@?"q@?((2r)!@r^r^-^1)^1^/^(^2^r^)@?f@?"p, where r=@?q/2@?. For the case q=4 and p=2, which is important in many applications, we prove a stronger inequality: @?f@?"4@?(2@r+1)^1^/^4@?f@?"2.