Lifting randomized query complexity to randomized communication complexity

Abstract: We show that for a relation $f\subseteq {0,1}^n\times \mathcal{O}$ and a function $$g:{0,1}^{m}\times {0,1}^{m} \rightarrow {0,1}$$ (with $m= O(\log n)$), $$\mathrm{R}<em>{1/3}(f\circ gⁿ⁾ = \Omega\left(\mathrm{R}</em>{1/3}(f) \cdot \left(\log\frac{1}{\mathrm{disc}(M_g)} - O(\log n)\right)\right),$$ where $f\circ g<sup>n$ represents the composition of $f$ and $g<sup>n$, $M_g$ is the sign matrix for $g$, $\mathrm{disc}(M_g)$ is the discrepancy of $M_g$ under the uniform distribution and $\mathrm{R}<em>{1/3}(f)$ ($\mathrm{R}</em>{1/3}(f\circ g<sup>n)$) denotes the randomized query complexity of $f$ (randomized communication complexity of $f\circ g<sup>n$) with worst case error $\frac{1}{3}$. In particular, this implies that for a relation $f\subseteq {0,1}^n\times \mathcal{O}$, $$\mathrm{R}<em>{1/3}(f\circ \mathrm{IP}_mⁿ⁾ = \Omega\left(\mathrm{R}</em>{1/3}(f) \cdot m\right),$$ where $$\mathrm{IP}_m:{0,1}^m\times {0,1}^m\rightarrow {0,1}$$ is the Inner Product (modulo $2$) function and $m= O(\log(n))$.