Difference between revisions of "Category:Monotone function"

Latest revision as of 18:18, 2 March 2020

For two vectors [math]x,y \in \{-1,1\}^n[/math], define a partial order relation by

[math]x \leq y \iff x_i \leq y_i ~~ \forall i = 1\ldots n.[/math]

A monotone Boolean function is a Boolean function [math]f:\{-1,1\}^n \to \{-1,1\}[/math] which is monotone in its input:

[math]x \leq y \Rightarrow f(x) \leq f(y) [/math].

The [math]i[/math]-th influence of a monotone function is equal to its first level Fourier coefficient: [math]\mathrm{Inf}_i(f) = \widehat{f}(\{i\})[/math].
For monotone functions, the total influence is bounded by the square root of the partition size: [math]\mathrm{Inf}(f) \leq \sqrt{\log(P(f))}[/math] ^[1].

TODO: Add properties about stability and noise sensitivity. See Mossel and O'Donnell.

The following 11 pages are in this category, out of 11 total.

@@ Line 9: / Line 9: @@
 ==Properties==
-* For monotone functions, the total [[influence]] is bounded by the square root of the [[partition size]]: <math>\mathrm{Inf}(f) \leq \sqrt{P(f)}</math> <ref>Ryan O'Donnell, Rocco Servedio, [https://www.cs.cmu.edu/~odonnell/papers/learn-monotone.pdf Learning Monotone Functions from Random Examples in Polynomial Time]</ref>.
+* The <math>i</math>-th [[influence]] of a monotone function is equal to its first level [[fourier representation | Fourier coefficient]]: <math>\mathrm{Inf}_i(f) = \widehat{f}(\{i\})</math>.
+* For monotone functions, the total [[influence]] is bounded by the square root of the [[partition size]]: <math>\mathrm{Inf}(f) \leq \sqrt{\log(P(f))}</math> <ref>Ryan O'Donnell, Rocco Servedio, [https://www.cs.cmu.edu/~odonnell/papers/learn-monotone.pdf Learning Monotone Functions from Random Examples in Polynomial Time]</ref>.
 * TODO: Add properties about stability and noise sensitivity. See Mossel and O'Donnell.