Let $\mathcal{F}_{n}^*$ be the set of Boolean functions depending on all $n$ variables. We prove that for any $f\in \mathcal{F}_{n}^*$, $f|_{x_i=0}$ or $f|_{x_i=1}$ depends on the remaining $n-1$ variables, for some variable $x_i$. This existent result suggests a possible way to deal with general Boolean functions via its subfunctions of some restrictions.
As an application, we consider the degree lower bound of representing polynomials over finite rings. Let $f\in \mathcal{F}_{n}^*$ and denote the exact representing degree over the ring $\mathbb{Z}_m$ (with the integer $m>2$) as $d_m(f)$. Let $m=Π_{i=1}^{r}p_i^{e_i}$, where $p_i$'s are distinct primes, and $r$ and $e_i$'s are positive integers. If $f$ is symmetric, then $m\cdot d_{p_1^{e_1}}(f)... d_{p_r^{e_r}}(f) > n$. If $f$ is non-symmetric, by the second moment method we prove almost always $m\cdot d_{p_1^{e_1}}(f)... d_{p_r^{e_r}}(f) > \lg{n}-1$. In particular, as $m=pq$ where $p$ and $q$ are arbitrary distinct primes, we have $d_p(f)d_q(f)=Ω(n)$ for symmetric $f$ and $d_p(f)d_q(f)=Ω(\lg{n}-1)$ almost always for non-symmetric $f$. Hence any $n$-variate symmetric Boolean function can have exact representing degree $o(\sqrt{n})$ in at most one finite field, and for non-symmetric functions, with $o(\sqrt{\lg{n}})$-degree in at most one finite field.
Optimal Bounds on Approximation of Submodular and XOS Functions by Juntas
IBM Research - Almaden（IBM阿尔马登研究中心）
We investigate the approximability of several classes of real-valued functions by functions of a small number of variables ({ \em juntas}). Our main results are tight bounds on the number of variables required to approximate a function $f:\{0,1\}^n \rightarrow [0,1]$ within $\ell_2$ -error $ε$ over the uniform distribution: 1. If $f$ is submodular, then it is $ε$ -close to a function of $O(\frac{1}{ε^2} \log \frac{1}ε)$ variables. This is an exponential improvement over previously known results. We note that $Ω(\frac{1}{ε^2})$ variables are necessary even for linear functions. 2. If $f$ is fra…
Approximate resilience, monotonicity, and the complexity of agnostic learning
University of Maryland（马里兰大学）；IBM Research - Almaden（IBM阿尔马登研究中心）；Columbia University（哥伦比亚大学）；Simons Institute, UC Berkeley（加州大学伯克利分校西蒙斯研究所）；Duquesne University（迪尤肯大学）
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…
