Difference between revisions of "Noise sensitivity"
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== Definition == | == 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>. | 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>. | ||
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| + | == 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) | ||
Revision as of 10:25, 23 September 2019
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)
