Difference between revisions of "Noise stability"
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* In case of <math>f:\{-1,1\}^{n}\rightarrow \{-1,1\}</math> , <math>Stab_{\rho}[f]=2\underset{\underset{\rho-correlated}{(x,y)}}{\mathbb{P}}[f(x)=f(y)]-1</math> | * In case of <math>f:\{-1,1\}^{n}\rightarrow \{-1,1\}</math> , <math>Stab_{\rho}[f]=2\underset{\underset{\rho-correlated}{(x,y)}}{\mathbb{P}}[f(x)=f(y)]-1</math> | ||
* For <math>f:\{-1,1\}^{n}\rightarrow \{-1,1\}</math>, and for <math>\rho∈[0,1]</math> the connection to [[Noise sensitivity]] is by <math>Stab_{\rho}[f]=1-2NS_{\frac{1}{2}-\frac{1}{2}\rho }[f]</math> | * For <math>f:\{-1,1\}^{n}\rightarrow \{-1,1\}</math>, and for <math>\rho∈[0,1]</math> the connection to [[Noise sensitivity]] is by <math>Stab_{\rho}[f]=1-2NS_{\frac{1}{2}-\frac{1}{2}\rho }[f]</math> | ||
| − | * The [[Majority]] function is assumed to be the least stable function among the [[Linear threshold]] when <math>n</math> is big and <math>\rho</math> is small. | + | * The [[Majority]] function is assumed to be the least stable function among the [[Linear threshold]] when <math>n</math> is big and <math>\rho</math> is small. <ref>Ryan O'Donnell, Analysis of Boolean functions, Chapter 2.4 [http://www.contrib.andrew.cmu.edu/~ryanod/?p=2245]</ref> |
== References == | == References == | ||
| − | < | + | <references/> |
Revision as of 09:49, 1 November 2020
Definition
For [math]f:\{-1,1\}^{n}\rightarrow\mathbb{\mathbb{R}} [/math] and [math]\rho∈[−1,1][/math], the noise stability of [math]f[/math] at [math]\rho[/math] is [math]Stab_{\rho}[f]=\underset{\underset{\rho-correlated}{(x,y)}}{\mathbb{E}}[f(x)f(y)][/math],
where [math] x [/math] and [math]y [/math] are [math]\rho[/math]-correlated if [math] y_{i}=\begin{cases}
x_{i} & with\space prabability\space \frac{1}{2}+\frac{1}{2}\rho\\
-x_{i} & with\space prabability\space \frac{1}{2}-\frac{1}{2}\rho
\end{cases}[/math]
Informally, the Stability tells us what is the chance for the function to stay the same when the input is under noise.
Properties
- In case of [math]f:\{-1,1\}^{n}\rightarrow \{-1,1\}[/math] , [math]Stab_{\rho}[f]=2\underset{\underset{\rho-correlated}{(x,y)}}{\mathbb{P}}[f(x)=f(y)]-1[/math]
- For [math]f:\{-1,1\}^{n}\rightarrow \{-1,1\}[/math], and for [math]\rho∈[0,1][/math] the connection to Noise sensitivity is by [math]Stab_{\rho}[f]=1-2NS_{\frac{1}{2}-\frac{1}{2}\rho }[f][/math]
- The Majority function is assumed to be the least stable function among the Linear threshold when [math]n[/math] is big and [math]\rho[/math] is small. [1]
