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en>Tomruen: /* Symmetry constructions */

2015-01-02T04:26:44Z

Symmetry constructions

← Older revision		Revision as of 06:26, 2 January 2015
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	~~In the [[mathematics\|mathematical]] discipline of [[graph theory]],~~ the '~~''expander walk sampling theorem''' states that [[Sampling (statistics)\|sampling]] [[vertex (graph theory)\|vertices]] in an [[expander graph]] by~~ doing a ~~[[Random walk#Random walk on graphs\|random walk]] is almost as good as sampling the vertices [[statistical independence\|independently]] from a [[uniform distribution (discrete)\|uniform distribution]]~~.		I am Oscar and I totally dig that title. His spouse doesn't like it the way he does but what he truly likes doing is to do aerobics and he's been doing it for quite a while. South Dakota is where me and my husband reside. My working day occupation is a librarian.<br><br>Look at my web page ... [http://www.sex-porn-tube.ch/user/JJeannere sex-porn-tube.ch]
	~~The earliest version of this theorem~~ is ~~due to {{harvtxt\|Ajtai\|Komlós\|Szemerédi\|1987}},~~ and ~~the more general version~~ is ~~typically attributed to {{harvtxt\|Gillman\|1998}}~~.

	~~==Statement==~~
	~~Let~~ <~~math~~>~~G = (V, E)~~<~~/math> be an expander graph with [[Expander graph#Spectral expansion\|normalized second-largest eigenvalue]] <math>\lambda</math~~>. ~~Let <math>n</math> denote the number of vertices in <math>G</math>~~. ~~Let <math>f : V \rightarrow [0, 1]</math> be a function on the vertices of <math>G</math>~~. ~~Let <math>\mu = E~~[f]</math> denote the true mean of <math>f</math>, i.e. <math>\mu = \frac{1}{n} \sum_{v \in V} f(v)</math>. Then, if we let <math>Y_0, Y_1, \ldots, Y_k</math> denote the vertices encountered in a <math>k</math>-step random walk on <math>G</math> starting at a random vertex <math>Y_0</math>, we have the following for all <math>\gamma > 0</math>:

	:~~<math>\Pr\left[\frac{1}{k} \sum_{i=0}^k f(Y_i) - \mu > \gamma\right] \leq e^{-\Omega (\gamma^2 (1-\lambda) k)}.</math>~~

	~~Here the <math>\Omega</math> hides an absolute constant <math>\geq 1~~/~~10<~~/~~math>~~. ~~An identical bound holds in the other direction:~~

	~~:<math>\Pr\left[\frac{1}{k} \sum_{i=0}^k f(Y_i) - \mu < -\gamma\right] \leq e^{-\Omega (\gamma^2 (1~~-~~\lambda) k)}.</math>~~

	~~==Uses==~~
	~~This theorem is useful in randomness reduction in the study of [[derandomization]]. Sampling from an expander walk is an example of a randomness~~-~~efficient [[sample (statistics)\|sampler]]~~. ~~Note that the number of [[bit]]s used in sampling <math>k<~~/~~math> independent samples from <math>f<~~/math> is <math>k \log n</math>, whereas if we sample from an infinite family of constant-degree expanders this costs only <math>\log n + O(k)</math>. Such families exist and are efficiently constructible, e.g. the [[Ramanujan graph]]s of [[Alexander Lubotzky\|Lubotzky]]-~~Phillips~~-~~Sarnak.~~

	~~==Notes==~~
	~~{{reflist}}~~

	~~==References==~~
	~~{{refbegin}}~~
	* {{citation
	~~\| first1=M. \| last1=Ajtai~~
	~~\| first2=J. \| last2=Komlós~~
	~~\| first3=E. \| last3=Szemerédi~~
	~~\| title=Deterministic simulation in LOGSPACE~~
	~~\| booktitle=Proceedings of the nineteenth annual ACM symposium on Theory of computing~~
	~~\| pages=132–140~~
	~~\| year=1987~~
	~~\| work=ACM~~
	~~\| doi=10.1145/28395.28410~~
	}}
	* {{citation
	~~\| first=D~~. ~~\| last=Gillman~~
	~~\| title=A Chernoff Bound for Random Walks on Expander Graphs~~
	~~\| journal=SIAM Journal on Computing~~
	~~\| volume=27~~
	~~\| pages=1203–1220~~
	~~\| year=1998~~
	~~\| publisher=Society for Industrial and Applied Mathematics~~
	~~\| doi=10.1137/S0097539794268765~~
	~~\| issue=4,~~
	}}
	~~{{refend}}~~

	~~==External links==~~
	* Proofs of the expander walk sampling theorem. [http://citeseer.ist.psu.edu/gillman98chernoff.html] [http://projecteuclid.org/Dienst/UI/1.0/Summarize/euclid.aoap/1028903453]

	~~[[Category:Sampling (statistics)]~~]

en>Tomruen: /* Symmetry constructions */

2013-11-22T05:53:21Z

Symmetry constructions

← Older revision		Revision as of 07:53, 22 November 2013
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	~~Greetings~~. Let ~~me start by telling you over the counter std test~~, [~~http~~://~~facehack~~.ir/~~index~~.~~php?do~~=/~~blog~~/20/~~what~~-~~you~~-~~have~~-to-do-~~facing~~-~~candida/ Suggested Resource site~~]~~, author's name~~ - ~~Phebe~~. ~~My day job~~ is ~~a meter reader~~. ~~What I adore doing~~ is ~~playing baseball but I haven't produced~~ a ~~dime with it~~. ~~South Dakota~~ is ~~exactly where me~~ and ~~my spouse live~~ and ~~my family enjoys it~~.		In the [[mathematics\|mathematical]] discipline of [[graph theory]], the '''expander walk sampling theorem''' states that [[Sampling (statistics)\|sampling]] [[vertex (graph theory)\|vertices]] in an [[expander graph]] by doing a [[Random walk#Random walk on graphs\|random walk]] is almost as good as sampling the vertices [[statistical independence\|independently]] from a [[uniform distribution (discrete)\|uniform distribution]].
			The earliest version of this theorem is due to {{harvtxt\|Ajtai\|Komlós\|Szemerédi\|1987}}, and the more general version is typically attributed to {{harvtxt\|Gillman\|1998}}.

			==Statement==
			Let <math>G = (V, E)</math> be an expander graph with [[Expander graph#Spectral expansion\|normalized second-largest eigenvalue]] <math>\lambda</math>. Let <math>n</math> denote the number of vertices in <math>G</math>. Let <math>f : V \rightarrow [0, 1]</math> be a function on the vertices of <math>G</math>. Let <math>\mu = E[f]</math> denote the true mean of <math>f</math>, i.e. <math>\mu = \frac{1}{n} \sum_{v \in V} f(v)</math>. Then, if we let <math>Y_0, Y_1, \ldots, Y_k</math> denote the vertices encountered in a <math>k</math>-step random walk on <math>G</math> starting at a random vertex <math>Y_0</math>, we have the following for all <math>\gamma > 0</math>:

			:<math>\Pr\left[\frac{1}{k} \sum_{i=0}^k f(Y_i) - \mu > \gamma\right] \leq e^{-\Omega (\gamma^2 (1-\lambda) k)}.</math>

			Here the <math>\Omega</math> hides an absolute constant <math>\geq 1/10</math>. An identical bound holds in the other direction:

			:<math>\Pr\left[\frac{1}{k} \sum_{i=0}^k f(Y_i) - \mu < -\gamma\right] \leq e^{-\Omega (\gamma^2 (1-\lambda) k)}.</math>

			==Uses==
			This theorem is useful in randomness reduction in the study of [[derandomization]]. Sampling from an expander walk is an example of a randomness-efficient [[sample (statistics)\|sampler]]. Note that the number of [[bit]]s used in sampling <math>k</math> independent samples from <math>f</math> is <math>k \log n</math>, whereas if we sample from an infinite family of constant-degree expanders this costs only <math>\log n + O(k)</math>. Such families exist and are efficiently constructible, e.g. the [[Ramanujan graph]]s of [[Alexander Lubotzky\|Lubotzky]]-Phillips-Sarnak.

			==Notes==
			{{reflist}}

			==References==
			{{refbegin}}
			* {{citation
			\| first1=M. \| last1=Ajtai
			\| first2=J. \| last2=Komlós
			\| first3=E. \| last3=Szemerédi
			\| title=Deterministic simulation in LOGSPACE
			\| booktitle=Proceedings of the nineteenth annual ACM symposium on Theory of computing
			\| pages=132–140
			\| year=1987
			\| work=ACM
			\| doi=10.1145/28395.28410
			}}
			* {{citation
			\| first=D. \| last=Gillman
			\| title=A Chernoff Bound for Random Walks on Expander Graphs
			\| journal=SIAM Journal on Computing
			\| volume=27
			\| pages=1203–1220
			\| year=1998
			\| publisher=Society for Industrial and Applied Mathematics
			\| doi=10.1137/S0097539794268765
			\| issue=4,
			}}
			{{refend}}

			==External links==
			* Proofs of the expander walk sampling theorem. [http://citeseer.ist.psu.edu/gillman98chernoff.html] [http://projecteuclid.org/Dienst/UI/1.0/Summarize/euclid.aoap/1028903453]

			[[Category:Sampling (statistics)]]

en>Tomruen: order

2012-06-26T21:39:08Z

order

New page

Greetings. Let me start by telling you over the counter std test, [http://facehack.ir/index.php?do=/blog/20/what-you-have-to-do-facing-candida/ Suggested Resource site], author's name - Phebe. My day job is a meter reader. What I adore doing is playing baseball but I haven't produced a dime with it. South Dakota is exactly where me and my spouse live and my family enjoys it.