Composition theorems in communication complexity

A well-studied class of functions in communication complexity are composed functions of the form $(f \comp g^n)(x,y)=f(g(x^1, y^1),..., g(x^n,y^n))$. This is a rich family of functions which encompasses many of the important examples in the literature. It is thus of great interest to understand what properties of $f$ and $g$ affect the communication complexity of $(f \comp g^n)$, and in what way. Recently, Sherstov \cite{She09b} and independently Shi-Zhu \cite{SZ09b} developed conditions on the inner function $g$ which imply that the quantum communication complexity of $f \comp g^n$ is at least the approximate polynomial degree of $f$. We generalize both of these frameworks. We show that the pattern matrix framework of Sherstov works whenever the inner function $g$ is {\em strongly balanced}---we say that $g: X \times Y \to \{-1,+1\}$ is strongly balanced if all rows and columns in the matrix $M_g=[g(x,y)]_{x,y}$ sum to zero. This result strictly generalizes the pattern matrix framework of Sherstov \cite{She09b}, which has been a very useful idea in a variety of settings \cite{She08b,RS08,Cha07,LS09,CA08,BHN09}. Shi-Zhu require that the inner function $g$ has small {\em spectral discrepancy}, a somewhat awkward condition to verify. We relax this to the usual notion of discrepancy. We also enhance the framework of composed functions studied so far by considering functions $F(x,y) = f(g(x,y))$, where the range of $g$ is a group $G$. When $G$ is Abelian, the analogue of the strongly balanced condition becomes a simple group invariance property of $g$. We are able to formulate a general lower bound on $F$ whenever $g$ satisfies this property.