Difference between revisions of "Category:Monotone function"
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==Properties== | ==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>. | ||
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* TODO: Add properties about stability and noise sensitivity. See Mossel and O'Donnell. | * TODO: Add properties about stability and noise sensitivity. See Mossel and O'Donnell. | ||
== References == | == References == | ||
<references/> | <references/> | ||
Revision as of 18:03, 2 March 2020
Definition
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].
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] [1].
- TODO: Add properties about stability and noise sensitivity. See Mossel and O'Donnell.
References
- ↑ Ryan O'Donnell, Rocco Servedio, Learning Monotone Functions from Random Examples in Polynomial Time
Pages in category "Monotone function"
The following 11 pages are in this category, out of 11 total.
