The estimates for Boolean surface area of polynomial threshold functions
Nankai University（南开大学）；Institute for Basic Science (IBS)（基础科学研究院）；University of Washington（华盛顿大学）；MSU（密歇根州立大学）；Max Planck Institute for Mathematics（马普数学研究所）；University of South Carolina（南卡罗来纳大学）
Fan Chang, Joseph Slote, Chun-Kai Tseng, Alexander Volberg, Haonan Zhang
Polynomial threshold functions (PTFs) are an important low-complexity class of Boolean functions, with strong connections to learning theory and approximation theory. Recent work on learning and testing PTFs has exploited structural and isoperimetric properties of the class, especially bounds on average sensitivity, one of the central themes in the study of PTFs since the Gotsman--Linial conjecture recently solved by OpenAI, \cite{GLsol}.
In this work we study PTFs through the lens of the Boolean surface area (or Talagrand boundary) \[ \mathbf{BSA}[f]=\mathbb{E}|\nabla f|=\mathbb{E}\sqrt{s_{f}(x)}, \] a natural measure of vertex-boundary complexity on the discrete cube. Our main result is that every degree-$d$ PTF has polylogarithmic Boolean surface area: \[ \mathbf{BSA}[f]\le C_d(\log(en))^{C_d}. \] The proof is based on the PTF Restriction Lemma of Kabanets, Kane, and Lu \cite{KKL2017} and proceeds through a tail bound for the pointwise sensitivity. In particular, it controls all subcritical fractional moments of the sensitivity. We also record a random block partition principle for Boolean surface area and an alternative recursive argument following Kane's work \cite{DK} on average sensitivity, which independently yields the second bound \[ \mathbf{BSA}[f]\le \exp(C(d)\sqrt{\log n}). \] This bound was obtained in \cite{FSVZ} without clarifying the magnitude of $C(d)$, which we do below. For lower moments $\bE |\nabla f|^η \le C(d, η)$ there is no dependence on $n$ for $0<η<1$.
Limits on representing Boolean functions by linear combinations of simple functions: thresholds, ReLUs, and low-degree polynomials
We consider the problem of representing Boolean functions exactly by "sparse" linear combinations (over $\mathbb{R}$ ) of functions from some "simple" class ${\cal C}$ . In particular, given ${\cal C}$ we are interested in finding low-complexity functions lacking sparse representations. When ${\cal C}$ is the set of PARITY functions or the set of conjunctions, this sort of problem has a well-understood answer, the problem becomes interesting when ${\cal C}$ is "overcomplete" and the set of functions is not linearly independent. We focus on the cases where ${\cal C}$ is the set of linear thresh…
Generating Boolean Functions on Totalistic Automata Networks
我们考虑研究有限状态机的任意网络的动态模拟能力的问题。在这些模型中，网络的每个节点都有两个状态0（被动）和1（主动）。节点的状态按照局部总体规则并行更新，即，仅取决于活动状态的总和。四个家庭的极权主义规则被认为是：线性或矩阵定义规则（如果其每个邻居都处于状态1，则节点采取状态1），阈值规则（如果其邻居的总和超过阈值，则节点采取状态1），孤立规则（如果相邻节点的和等于某个单个数字，则节点采取状态1）和区间规则（如果其邻居的和属于某个离散区间，则节点采取状态1）。我们专注于研究后一类的动态模拟能力。特别是，我们证明了由矩阵定义的规则所控制的整体自动机网络只能实现常数函数和其他矩阵定义的函数。此外，我们还证明了t by threshold规则可以生成任意单调布尔函数。最后，我们证明了由孤立和区间规则驱动的网络表现出非常丰富的布尔函数谱，因为它们实际上可以实现任何任意布尔函数。我们补充这一结果通过实验研究随机图上的总体规则所产生的一组不同的布尔函数。
Improved Approximation of Linear Threshold Functions
We prove two main results on how arbitrary linear threshold functions $f(x) = \sign(w\cdot x - θ)$ over the $n$ -dimensional Boolean hypercube can be approximated by simple threshold functions. Our first result shows that every $n$ -variable threshold function $f$ is $\eps$ -close to a threshold function depending only on $\Inf(f)^2 \cdot \poly(1/\eps)$ many variables, where $\Inf(f)$ denotes the total influence or average sensitivity of $f.$ This is an exponential sharpening of Friedgut's well-known theorem \cite{Friedgut:98} , which states that every Boolean function $f$ is $\eps$ -close to …
