For a sequence of Boolean functions $f_n : \{-1,1\}^{V_n} \longrightarrow \{-1,1\}$, defined on increasing configuration spaces of random inputs, we say that there is sparse reconstruction if there is a sequence of subsets $U_n \subseteq V_n$ of the coordinates satisfying $|U_n| = o(|V_n|)$ such that knowing the coordinates in $U_n$ gives us a non-vanishing amount of information about the value of $f_n$. We first show that, if the underlying measure is a product measure, then no sparse reconstruction is possible for any sequence of transitive functions. We discuss the question in different frameworks, measuring information content in $L^2$ and with entropy. We also highlight some interesting connections with cooperative game theory. Beyond transitive functions, we show that the left-right crossing event for critical planar percolation on the square lattice does not admit sparse reconstruction either. Some of these results answer questions posed by Itai Benjamini.

翻译：对于定义在递增随机输入配置空间上的布尔函数序列$f_n : \{-1,1\}^{V_n} \longrightarrow \{-1,1\}$，若存在坐标子集序列$U_n \subseteq V_n$满足$|U_n| = o(|V_n|)$，且已知$U_n$中的坐标能提供关于$f_n$值的非消失信息量，则称存在稀疏重构。我们首先证明：若底层测度为乘积测度，则任何传递函数序列均不可能实现稀疏重构。我们从$L^2$范数和熵两个不同框架讨论信息含量的度量问题，并揭示其与合作博弈论的有趣联系。除传递函数外，我们还证明正方形网格上临界平面渗透的左-右穿越事件同样不允许稀疏重构。部分结果回答了Itai Benjamini提出的问题。