Difference between revisions of "Fourier representation"
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where <math>\hat{f}_S</math> are real numbers called the '''Fourier coefficients''' of <math>f</math>. | where <math>\hat{f}_S</math> are real numbers called the '''Fourier coefficients''' of <math>f</math>. | ||
| − | The set of functions <math>\{\underset{i\in S}{\prod}x_{i}|S\subseteq | + | The set of functions <math>\{\underset{i\in S}{\prod}x_{i}|S\subseteq [n] \}</math> is an orthonormal basis of the vector space of all boolean functions <math>f:\{-1,1\}^{n}\to\mathbb{R}</math> with the inner product:<br><br> |
| − | <math><f,g>=2^{-n}\underset{x\in\{-1,1\}^{n}}{\sum}f(x)g(x)=\underset{x\sim\{-1,1\}^{n}}{\mathbb{E}}[f(x)g(x)]</math><br> | + | <math>\left<f,g\right>=2^{-n}\underset{x\in\{-1,1\}^{n}}{\sum}f(x)g(x)=\underset{x\sim\{-1,1\}^{n}}{\mathbb{E}}[f(x)g(x)].</math><br> |
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| − | + | === The Fourier Weight === | |
| − | <math> | + | For <math>f:\{-1,1\}^{n}\to\mathbb{R}</math> and <math>0\leq k\leq n</math> the Fourier weight of <math>f</math> at degree <math>k</math> (or depth <math>k</math>) is: |
| − | + | <math>W^{k}[f]=\underset{S\subseteq\{1,...n\},|s|=k}{\sum\hat{f}(S)^{2}}</math> | |
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| − | + | === Some useful identities === | |
| − | <math> | + | Parseval's identity: <math>\left<f,f\right>=\underset{S\in [n]}{\sum}\hat{f}(S)^{2}</math> |
| − | + | Plancharel's identity: <math>\left<f,g\right>=\underset{S\in [n]}{\sum}\hat{f}(S)\hat{g}(S)</math> | |
| − | + | The convolution theorem: <math>\widehat{f * g}(S) = \widehat{f}(S) \widehat{g}(S)</math>, where <math>(f*g)(x) = \mathbb{E}_{y}\left[f(y)g(x \circ y) \right]</math> (where <math>\circ</math> is bit-wise multiplication). | |
== Properties == | == Properties == | ||
Latest revision as of 09:13, 15 November 2019
General
Every Boolean function [math]f:\{-1,1\}^n \to \{-1,1\}[/math] may be uniquely written as a multivariate polynomial:
[math] f(x) = \sum_{S \subseteq [n]} \hat{f}_S \prod_{i \in S} x_i [/math],
where [math]\hat{f}_S[/math] are real numbers called the Fourier coefficients of [math]f[/math].
The set of functions [math]\{\underset{i\in S}{\prod}x_{i}|S\subseteq [n] \}[/math] is an orthonormal basis of the vector space of all boolean functions [math]f:\{-1,1\}^{n}\to\mathbb{R}[/math] with the inner product:
[math]\left\lt f,g\right\gt =2^{-n}\underset{x\in\{-1,1\}^{n}}{\sum}f(x)g(x)=\underset{x\sim\{-1,1\}^{n}}{\mathbb{E}}[f(x)g(x)].[/math]
The Fourier Weight
For [math]f:\{-1,1\}^{n}\to\mathbb{R}[/math] and [math]0\leq k\leq n[/math] the Fourier weight of [math]f[/math] at degree [math]k[/math] (or depth [math]k[/math]) is:
[math]W^{k}[f]=\underset{S\subseteq\{1,...n\},|s|=k}{\sum\hat{f}(S)^{2}}[/math]
Some useful identities
Parseval's identity: [math]\left\lt f,f\right\gt =\underset{S\in [n]}{\sum}\hat{f}(S)^{2}[/math]
Plancharel's identity: [math]\left\lt f,g\right\gt =\underset{S\in [n]}{\sum}\hat{f}(S)\hat{g}(S)[/math]
The convolution theorem: [math]\widehat{f * g}(S) = \widehat{f}(S) \widehat{g}(S)[/math], where [math](f*g)(x) = \mathbb{E}_{y}\left[f(y)g(x \circ y) \right][/math] (where [math]\circ[/math] is bit-wise multiplication).
Properties
- TODO. Start off with relation between Fourier weight and other complexity measures such as circuit size, sensitivity, decision tree depth.
