We prove a sensitivity-to-communication lifting theorem for arbitrary gadgets. Given functions $f: \{0,1\}^n\to \{0,1\}$ and $g : \mathcal X\times \mathcal Y\to \{0,1\}$, denote $f\circ g(x,y) := f(g(x_1,y_1),\ldots,g(x_n,y_n))$. We show that for any $f$ with sensitivity $s$ and any $g$, \[D(f\circ g) \geq s\cdot \bigg(\frac{\Omega(D(g))}{\log\mathsf{rk}(g)} - \log\mathsf{rk}(g)\bigg),\] where $D(\cdot)$ denotes the deterministic communication complexity and $\mathsf{rk}(g)$ is the rank of the matrix associated with $g$. As a corollary, we get that if $D(g)$ is a sufficiently large constant, $D(f\circ g) = \Omega(\min\{s,d\}\cdot \sqrt{D(g)})$, where $s$ and $d$ denote the sensitivity and degree of $f$. In particular, computing the OR of $n$ copies of $g$ requires $\Omega(n\cdot\sqrt{D(g)})$ bits.

翻译：我们证明了针对任意门电路的灵敏度-通信复杂度提升定理。给定函数 $f: \{0,1\}^n\to \{0,1\}$ 和 $g : \mathcal X\times \mathcal Y\to \{0,1\}$，记 $f\circ g(x,y) := f(g(x_1,y_1),\ldots,g(x_n,y_n))$。我们证明，对于任意灵敏度为 $s$ 的函数 $f$ 和任意函数 $g$，有 \[D(f\circ g) \geq s\cdot \bigg(\frac{\Omega(D(g))}{\log\mathsf{rk}(g)} - \log\mathsf{rk}(g)\bigg),\] 其中 $D(\cdot)$ 表示确定性通信复杂度，$\mathsf{rk}(g)$ 是与 $g$ 关联的矩阵的秩。作为推论，我们得到：若 $D(g)$ 是一个足够大的常数，则 $D(f\circ g) = \Omega(\min\{s,d\}\cdot \sqrt{D(g)})$，其中 $s$ 和 $d$ 分别表示 $f$ 的灵敏度和次数。特别地，计算 $n$ 个 $g$ 副本的 OR 运算需要 $\Omega(n\cdot\sqrt{D(g)})$ 比特的通信量。