$ \newcommand{\undefined}{} \newcommand{\hfill}{} \newcommand{\qedhere}{\square} \newcommand{\qed}{\square} \newcommand{\ensuremath}[1]{#1} \newcommand{\bit}{\{0,1\}} \newcommand{\Bit}{\{-1,1\}} \newcommand{\Stab}{\mathbf{Stab}} \newcommand{\NS}{\mathbf{NS}} \newcommand{\ba}{\mathbf{a}} \newcommand{\bc}{\mathbf{c}} \newcommand{\bd}{\mathbf{d}} \newcommand{\be}{\mathbf{e}} \newcommand{\bh}{\mathbf{h}} \newcommand{\br}{\mathbf{r}} \newcommand{\bs}{\mathbf{s}} \newcommand{\bx}{\mathbf{x}} \newcommand{\by}{\mathbf{y}} \newcommand{\bz}{\mathbf{z}} \newcommand{\Var}{\mathbf{Var}} \newcommand{\dist}{\text{dist}} \newcommand{\norm}[1]{\\|#1\\|} \newcommand{\etal} \newcommand{\ie} \newcommand{\eg} \newcommand{\cf} \newcommand{\rank}{\text{rank}} \newcommand{\tr}{\text{tr}} \newcommand{\mor}{\text{Mor}} \newcommand{\hom}{\text{Hom}} \newcommand{\id}{\text{id}} \newcommand{\obj}{\text{obj}} \newcommand{\pr}{\text{pr}} \newcommand{\ker}{\text{ker}} \newcommand{\coker}{\text{coker}} 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Back to Boolean Function Analysis
Back to notes

Some definitions and useful facts

50%

If not specify, n the following, we let $f:\Bit^n\rightarrow\Bit$ be arbitrary boolean function and $\mathbf{f}:\Bit^n\rightarrow\Bit$ be a random boolean function in the sense that for any $x\in\Bit^n$, the probability of $\mathbf{f}(x)=1$ is half.

Basic Fourier analysis of boolean function

Let $S\subseteq[n]$, define the character for $S$ as $\chi_S:\Bit^n\rightarrow\Bit$ such that $\chi_S(x)=\prod_{i\in S}x_i$.

$\{\chi_S\}_{S\subseteq[n]}$ forms an orthonormal basis for $\mathcal{F}_n$, i.e., $\langle\chi_S,\chi_T\rangle=\mathbf{1}_{S=T}$

Let $f:\Bit^n\rightarrow\R$, define the Fourier expansion of $f$ as follows. \begin{equation} f = \sum_{S\subseteq[n]}\widehat{f}(S)\chi_S, \end{equation} where we call $\widehat{f}(S)$ the Fourier coefficient of $f$.

  • $\widehat{f}(S) = \langle f,\chi_S\rangle$
  • (Parseval) $\langle f,f\rangle = \sum_{S\subseteq[n]}\widehat{f}(S)^2$
  • (Plancherel) $\langle f,g\rangle = \sum_{S\subseteq[n]}\widehat{f}(S)\widehat{g}(S)$
  • $\Var[f] = \sum_{S\subseteq[n],S\neq\emptyset}\widehat{f}(S)^2$
  • $\dist(f,g) := \mathbb{P}_{\bx}[f(\bx)\neq g(\bx)]$
    • $\langle f,g\rangle = 1-2\dist(f,g)$
    • $f$ is $\left(\max_{S\subseteq[n]}\widehat{f}(S)\right)$-close to linear

The convolution of $f$ and $g$ is $f\star g(x):=\mathbb{E}_{\by}[f(x-\by)g(\by)]$

  • $f\star g(x):=\mathbb{E}_{\by}[f(x+\pm\by)g(\by)]=\mathbb{E}_{\by}[f(\by)g(x\pm\by)]$
  • $\widehat{f\star g}(S) = \widehat{f}(S)\cdot\widehat{g}(S)$

Influence, noise stability, and noise sensibility

For any $i\in[n]$, define the $i$th influence of $f$ as \begin{equation} \bfI_i[f] := \mathbb{P}[f(x_1,\dots,x_{i-1},0,x_{i+1},\dots,x_n)\neq f(x_1,\dots,x_{i-1},1,x_{i+1},\dots,x_n)]. \end{equation} The total influence of $f$ is defined as the sum of coordinate-wise influence, i.e., \begin{equation} \bfI[f] := \sum_{i\in[n]}\bfI_i[f]. \end{equation}

  • $\bfI_i[f] = \sum_{S:i\in S}\widehat{f}(S)^2$
  • If $f$ is monotone, then $\bfI_i[f]=\widehat{f}(\{i\})$
  • $\bfI[f]=\sum_{k\in[n]}k\cdot \mathbf{W}^k[f]$
  • $\bfI[f]=2\mathbb{E}_{x\sim\Bit^n}[\sharp(-1)\text{-pivotal coordinates for }f\text{ on }x]$
  • $\bfI_i[f]=\mathbb{E}_{x\sim\Bit^n}[(D_if)^2]$
  • $\bfI[f]=\langle f,Lf\rangle=\langle Lf,Lf\rangle$
  • (Poincare Inequality) For unbiased $f$, $\bfI[f]\geq1$.
  • (KKL Theorem) $\max_{i\in[n]}{\bfI_i[f]}\geq\Var[f]\cdot\Omega(\frac{\log n}{n})$
Function Influence
$\text{Maj}_n$ $\Theta(\sqrt{n})$
DNF/CNF of width $w$ $\leq2w$
DNF/CNF of size $s$ $\leq O(\log s)$
$\text{Tribe}_n$ $=(\ln n)(1\pm o(1))$

Let $\rho\in[-1,1]$ and $x\in\Bit^n$, we say random vector $\mathbf{y}\in\Bit^n$ is $\rho$-correlated with $x$ for some $\rho\in[0,1]$ if for each $i\in[n]$, \(\mathbf{y}_i=\left\{\begin{array} \sign(\rho)x_i&,\text{w.p. }\card{\rho}\\ \text{uniformly random}&,\text{w.p. }1-\card{\rho} \end{array}\right.\) We denote it as $\by\sim N_{\rho} x$.

Furthermore, when we draw $\bx$ uniformly from $\Bit^n$ and $\by\sim N_{\rho}(\bx)$, we call $(\bx,\by)$ a $\rho$-correlation pair.

  • $\mathbb{E}[\bx_i]\mathbb{E}[\by_i]=0$
  • $\mathbb{E}[\bx_i\by_i]=\rho$

Define the noise stability of $f$ as $\Stab[f]:=\mathbb{E}_{\bx,\by\sim_{\rho}x}[f(\bx)f(\by)]$

  • $\Stab_{\rho}[\chi_S]=\rho^{\card{S}}$
  • $\Stab_{\rho}[f] = \sum_{k\in[n]}\rho^k\bfW^k[f]$
  • $\bfI = \frac{d}{d\rho}\Stab_{\rho}[f]|_{\rho=1}$

Let $x\in\Bit^n$, we say random vector $\mathbf{y}\in\Bit^n$ is yielded by $x$ with flipping probability $\delta$ for some $\delta\in[0,1]$ if for each $i\in[n]$, \(\mathbf{y}_i=\left\{\begin{array} xx_i&,\text{w.p. }1-\delta\\ -x_i&.\text{w.p. }\delta \end{array}\right.\) We denote it as $\by\leftarrow_{\delta} x$.

  • When $\delta=\frac{1}{2}-\frac{1}{2}\rho$, a.k.a., $\rho=1-2\delta$, $\bx\leftarrow_{\delta}\by$ is equivalent to $(\bx,\by)$ being a $\rho$-correlation pair.

Define the noise sensitivity of $f$ with flipping probability $\delta$ as \begin{equation} \NS_{\delta}[f] := \mathbb{P}_{\bx,\by\leftarrow_{\delta}\bx}[f(\bx)\neq f(\by)]. \end{equation}

  • $\NS_{\delta}[f] = \frac{1}{2}-\frac{1}{2}\Stab_{1-2\delta}[f]$
  • $\NS_{\delta}[f] = \frac{1}{2}\sum_{0\leq k\leq n}(1-(1-2\delta)^k)\cdot \bfW^k[f]$

For any $\rho\in[-1,1]$, define operator $T_{\rho}$ as follows. \begin{equation} T_{\rho}f(x) := \mathbb{E}_{\by\sim N_{\rho}(x)}[f(\by)]. \end{equation}

  • $T_{\rho}f = \sum_{S\subseteq[n]}\rho^{\card{S}}\widehat{f}(S)\chi_S$
  • $\Stab_{\rho}[f]=\langle f,T_{\rho}f \rangle$

Inspired by the fact that $\bfI_i[f]=\mathbb{E}_{x\sim\Bit^n}[(D_if)^2]$, we define the following notion of stable influence to connect $\bfI$ and $\Stab$.

Let $\rho\in[-1,1]$, define the $\rho$-stable influence of $f$ as \begin{equation} \bfI_i^{(\rho)}[f] := \Stab_{\rho}[D_if] = \sum_{i\in S}\rho^{\card{S}-1}\widehat{f}(S)^2. \end{equation} Similarly, define $\bfI^{(\rho)}[f] = \sum_{i\in[n]}\bfI_i^{(\rho)}[f]$.

  • Let $f:\Bit^n\rightarrow\R$ such that $\Var[f]\leq1$ and $0<\delta,\epsilon<1$, we have $\card{{i\in[n]:\ \bfI_i^{(1-\delta)}[f]\geq\epsilon}}\leq\frac{1}{\delta\epsilon}$. (See this note for the proof of a key inequality)

Spectral properties

For any $\epsilon>0$, we say a function $f:{-1,1}^n\rightarrow\mathbb{R}$ is $\epsilon$-concentrated on degree up to $k$ if $\mathbf{W}^{\geq k}[f]\leq\epsilon$.

Random function

In the following, let $f:\{-1,1\}^n\rightarrow\mathbb{R}$ be a random boolean function in the sense that for any $x\in\{-1,1\}^n$, the probability of $f(x)=1$ is half.