一个非近似Junta的低深度单调函数

对于浅单调电路，加法比计数更难

Addition is exponentially harder than counting for shallow monotone circuits

令$U_{k,N}$表示布尔函数，该布尔函数将每个$N$位的$k$串作为输入，表示$\{0,1,\dots,2^{N}-1\}$中的$k$号$a^{(1)},\dots,a^{(k)}$，并且当且仅当$a^{(1)} + \cdots + a^{(k)} \geq 2^N.$时输出1。令THR $_{t,n}$表示单调未加权阈值门，即，布尔函数，它将单个字符串$x \in \{0,1\}^n$作为输入，当且仅当$x_1 + \cdots + x_n \geq t$时输出$1$。我们把由THR门组成的电路称为单调多数电路。本文的主要结果是计算$U_{k,N}$的有界深度单调多数电路的大小的指数下界。更准确地说，我们表明，任何常数$d \geq 2$，任何深度$d$单调多数电路计算$U_{d,N}$必须有大小$\smash{2^{Ω(N^{1/d})}}$。由于$U_{k,N}$可以通过单个单调加权阈值门（使用指数大的权重）来计算，因此我们的下限意味着恒定深度单调多数电路需要指数大小来模拟单调加权阈值门。这回答了Goldmann和Karpinski（STOC'93）提出的一个问题，最近由Hastad（2010，2014）重申。我们还表明，我们的下限基本上是最好的可能，通过构建一个深度- $d$，大小- $2^{O(N^{1/d})}$单调多数电路$U_{d,N}$。作为我们的下限的推论…

同态多项式的单调有界深度复杂度

Monotone Bounded-Depth Complexity of Homomorphism Polynomials

本文研究了同态多项式和彩色子图多项式的单调有界深度电路复杂度，提出图参数层次$\mathrm{tw}_Δ(H)$并证明了最优深度层次定理。

关于单调ReLU神经网络与ICNN的深度

On the Depth of Monotone ReLU Neural Networks and ICNNs

本文研究单调ReLU网络和输入凸神经网络（ICNN）的表达能力，证明了单调网络无法计算或近似最大值函数，并给出了ICNN计算该函数的精确深度下界，同时证明了ReLU网络与ICNN之间的深度分离。

近似弹性，单调性和不可知学习的复杂性

Approximate resilience, monotonicity, and the complexity of agnostic learning

A function $f$ is $d$ -resilient if all its Fourier coefficients of degree at most $d$ are zero, i.e., $f$ is uncorrelated with all low-degree parities. We study the notion of $\mathit{approximate}$ $\mathit{resilience}$ of Boolean functions, where we say that $f$ is $α$ -approximately $d$ -resilient if $f$ is $α$ -close to a $[-1,1]$ -valued $d$ -resilient function in $\ell_1$ distance. We show that approximate resilience essentially characterizes the complexity of agnostic learning of a concept class $C$ over the uniform distribution. Roughly speaking, if all functions in a class $C$ are far…

Karchmer-Wigderson Games for Hazard-free Computation

We present a Karchmer-Wigderson game to study the complexity of hazard-free formulas. This new game is both a generalization of the monotone Karchmer-Wigderson game and an analog of the classical Boolean Karchmer-Wigderson game. Therefore, it acts as a bridge between the existing monotone and general games. Using this game, we prove hazard-free formula size and depth lower bounds that are provably stronger than those possible by the standard technique of transferring results from monotone complexity in a black-box fashion. For the multiplexer function we give (1) a hazard-free formula of optim…

Approximating the Distance to Monotonicity of Boolean Functions

We design a nonadaptive algorithm that, given oracle access to a function $f: \{0,1\}^n \to \{0,1\}$ which is $α$-far from monotone, makes poly$(n, 1/α)$ queries and returns an estimate that, with high probability, is an $\widetilde{O}(\sqrt{n})$-approximation to the distance of $f$ to monotonicity. The analysis of our algorithm relies on an improvement to the directed isoperimetric inequality of Khot, Minzer, and Safra (SIAM J. Comput., 2018). Furthermore, we rule out a poly$(n, 1/α)$-query nonadaptive algorithm that approximates the distance to monotonicity significantly better by showing th…