Difference between revisions of "Noise stability"
Or elmackias (talk | contribs) (Created page with " == Definition == f:{− 1 , 1 } n → R and ρ ∈ [ − 1 , 1], the noise stability of f at ρ is Stab ρ [f] = E[f(x)f(y)]. ( x,y ) ρ -correlated") |
Or elmackias (talk | contribs) (→Definition) |
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== Definition == | == 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> | ||
Revision as of 07:41, 23 September 2019
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]
