Noise sensitivity

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Definition

For [math]f:\{-1,1\}^{n}\longrightarrow\{-1,1\}[/math] and [math]\delta\in[0,1][/math], the noise sensitivity of [math]f[/math] at [math]\delta[/math], [math]NS_{\delta}[f][/math], is the probability that [math]f(x)\neq f(y)[/math] when [math]x\sim\{-1,1\}^{n}[/math] is uniformly random and [math]y[/math] is formed from [math]x[/math] by reversing each bit independently with probability [math]\delta[/math].

Properties

  • The connection between noise sensitivity and Stability is given by [math]NS_{\delta}[f]=\frac{1}{2}-\frac{1}{2}Stab_{1-2\delta}[f][/math]
  • For [math]\delta\leq\frac{1}{2}[/math], and [math]f[/math] is Linear threshold function ([math]f(x)=sgn(\sum_{i=1}^{n}\omega_{i}x_{i}-t)[/math]), then [math]NS_{\delta}[f]\leq2\delta^{1/2}[/math]. furthermore, [math]\underset{\delta\rightarrow0}{\limsup}\frac{\limsup_{n\rightarrow\infty}\sup_{\omega,t}NS_{\delta}[f]}{\sqrt{\delta}}\leq\sqrt{\frac{2}{\pi}}[/math] (Peres, 2006)

References

[1]

[2]

  1. Ryan O'Donnell, Analysis of Boolean functions, Chapter 2.4 [1]
  2. Peres (2004). "Noise Stability of Weighted Majority" [2]