Heat kernel signature - Revision history

en>Myasuda: /* Time parameter selection */ sp

2014-11-22T15:50:56Z

Time parameter selection: sp

← Older revision		Revision as of 17:50, 22 November 2014
Line 1:		Line 1:
	~~The '''MM algorithm'''~~ is an iterative [[optimization]] method which exploits the [[convex function\|convexity]] of a function in order to find their maxima or minima. The '''MM''' stands for “Majorize-Minimization” or “Minorize-Maximization”, depending on whether you’re doing maximization or minimization. '''MM''' itself is not an algorithm, but a description of how to construct an [[optimization algorithm]].		Jason is his name and he fully digs that title. Taking part in [http://browse.deviantart.com/?qh=&section=&global=1&q=basketball basketball] is a person of the matters he loves most. Alaska has constantly been his home. Invoicing is where by his main profits arrives from. He is not godd at design and style but you might want to examine his internet site: http://www.benillup.cat/aspnet_client/salomon-fusion-05586477.htm<br><br>Feel free to visit my website ... [http://www.benillup.cat/aspnet_client/salomon-fusion-05586477.htm salomon fusion]

	~~The [[EM algorithm]] can be treated as a special case of the MM algorithm. However, in the EM algorithm complex [[conditional expectation]]~~ and ~~extensive analytical skills are usually involved, while in the MM algorithm convexity and inequalities are our major focus, and it is relatively easier to understand and apply in most of the cases.~~

	~~==History==~~
	~~The original idea of the '''MM algorithm''' can be dated back at least to 1970 when Ortega and Rheinboldt were doing their studies related to [[line search]] methods~~.~~<ref>~~
	~~{{cite journal~~
	~~\|last1=Ortega \|first1=J.M.~~
	~~\|last2=Rheinboldt\|first2=W.C.~~
	~~\|title=Iterative Solutions of Nonlinear Equations~~ in ~~Several Variables~~
	~~\|journal=[~~[~~New York~~: ~~Academic]]~~
	~~\|year=1970 \|pages=253–255~~
	}}
	</~~ref> The same idea kept reappearing under different guise in different areas since, until 2000 Hunter and Lange put all in to a general frame works and named '''MM''' for the first time~~.~~<ref>~~
	~~{{cite journal~~
	~~\|last1=Hunter\|first1=D~~.R.
	~~\|last2~~=~~Lange\|first2~~=K.
	~~\|title~~=~~Quantile Regression via an MM Algorithm~~
	~~\|journal~~=~~[[Journal of Computational~~
	~~and Graphical Statistics~~]]
	~~\|year=2000 \|volume=9 \|pages=60–77~~
	}}
	~~</ref> Recently studies have shown that it can be used in~~ a ~~wide range~~ of ~~context, like [[mathematics]], [[statistics]], [[machine learning]], [[engineering]], etc~~.

	~~==How it works==~~
	~~'''MM algorithm''' works~~ by ~~finding a surrogate function that minorizes or majorizes the objective function~~. ~~Optimizing the surrogate functions will drive the objective function upward or downward until a local [[optimum]]~~ is ~~reached.~~

	~~Take the '''minorize-maximazation''' version for example.~~

	~~Let <math> f(\theta) </math> be the objective convex function we~~ want to ~~maximize. At the <math> m <~~/~~math> step of the algorithm, <math> m=0,1~~..~~. <~~/~~math>, the constructed function <math> g(\theta\|\theta_m) <~~/~~math> will be called the minorized version of the objective function (the surrogate function) at~~ <~~math~~> ~~\theta_m~~ <~~/math~~> if

	~~<math> g(\theta\|\theta_m)</math> ≤ <math> f(\theta) </math> for all <math> \theta </math>~~
	~~<math> g(\theta_m\|\theta_m)=f(\theta_m) </math>~~

	~~Then we maximize <math> g(\theta\|\theta_m) </math> instead of <math> f(\theta) </math>, and let~~

	~~<math> \theta_{m+1}=\arg\max_{\theta}g(\theta\|\theta_m) </math>~~

	~~The above iterative method will guarantee that <math> f(\theta_m) </math> will converge~~ to ~~a local optimum or a saddle point as <math> m <~~/~~math> goes to infinity, because by the construction we have~~
	~~<math> f(\theta_{m+1})<~~/~~math> ≥ <math> g(\theta_{m+1}\|\theta_m)<~~/~~math> ≥ <math> g(\theta_m\|\theta_m)= f(\theta_m)<~~/~~math>~~

	~~The marching of <math>\theta_m </math> and the surrogate functions relative to the objective function is shown on the Figure [[File:Mmalgorithm.jpg\|right\|thumb\|MM algorithm]]~~

	~~We can just flip the image upside down, and that would be the methodology while we are doing '''Majorize~~-~~Minimization'''~~.

	~~==Ways to construct surrogate functions==~~
	~~Basically, we can use any inequalities to construct the desired majorized/minorized version of the objective function, but there are several typical choices~~
	* [[Jensen's inequality]]
	* [[Convexity inequality]]
	* [[Cauchy–Schwarz inequality]]
	* [[Inequality of arithmetic and geometric means]]

	~~==References==~~
	~~{{reflist}}~~

	~~[[Category:Optimization algorithms and methods]~~]

en>Addbot: Bot: Removing Orphan Tag (Nolonger an Orphan) (Report Errors)

2013-01-09T01:21:37Z

Bot: Removing Orphan Tag (Nolonger an Orphan) (Report Errors)

← Older revision		Revision as of 03:21, 9 January 2013
Line 1:		Line 1:
	~~Back pain~~ is ~~anything that can be a really debilitating affliction for numerous. The crucial to conquering~~ the ~~inconvenience and agony~~ of ~~this condition is~~ to ~~learn as significantly as attainable about its causes and therapies~~. ~~Put the suggestions and info in this write~~-~~up to function~~, ~~and you will stand~~ a ~~very good possibility~~ of ~~significantly increasing your high quality of life~~.~~<br><br>~~		The '''MM algorithm''' is an iterative [[optimization]] method which exploits the [[convex function\|convexity]] of a function in order to find their maxima or minima. The '''MM''' stands for “Majorize-Minimization” or “Minorize-Maximization”, depending on whether you’re doing maximization or minimization. '''MM''' itself is not an algorithm, but a description of how to construct an [[optimization algorithm]].

	~~Use your legs anytime you lift anything~~. ~~You have to have a powerful base with your legs~~ and ~~your physique requirements to be even. Hold~~ the ~~object you~~ are ~~lifting up close~~ to ~~you as you lift from your legs~~. ~~This will help avoid any back injury that may well happen.<br><br>If you read for extended amounts~~ of ~~time every single day - either for private pleasure or specialist factors - then you~~ can ~~keep away from~~ back ~~pain resulting from neck strain by keeping your head level and bringing the documents up to that level. Keeping your head bent or raised~~ at ~~unnatural angles for extended amounts of time can lead~~ to ~~strain. So getting a document hanger or holding your materials up as an alternative of setting them on a desk or in your lap can support~~ to ~~avoid cumulative neck injuries associated to these posture no-no"s~~.<br>~~<br>Finding out to lift correctly can save oneself a lot of back pain in the future~~. ~~Understanding to lift from the knees instead of just bending over to lift will tremendously reduce the strain on your back~~. ~~This will maintain your back from being over exerted and assist you keep your back from establishing pain earlier~~.<br>~~<br>Getting back pain? Get a massage. Acquiring a back massage will ease the sore tensed muscle tissues~~ in ~~your back~~, and ~~aid~~ to ~~relieve~~ the ~~tension of back pain. Possessing a 30 minute massage either by a professional or a loved ones member can create lengthy lasting relief from sore back muscle tissues~~.<br>~~<br>No matter what the explanation, if you suffer from back discomfort and you have to bend over, be sure to do so with your knees and not your back~~. ~~Learn further on the affiliated essay by visiting [http://guzmanyeet~~.~~wordpress~~.~~com All Diva Media Blog~~ \| ~~Felix Salmon - A Slice~~ of ~~Lime in the Soda~~]~~. A lot of men and women endure from back strains or pains due to the fact they bend over utilizing their back, which puts also much stress on the spine.~~<br>~~<br>Chronic muscle pain, such as back pain,~~ can be ~~brought on by~~ a ~~vitamin D deficiency. To get your fair share~~ of ~~this vitamin~~, ~~consume lots of tiny-boned fish~~, ~~fortified milk and cereal. Also~~, ~~be positive to get frequent exposure to sunlight and do not forget to use that sunscreen!<br><br>If you endure from chronic back discomfort~~, ~~getting~~ a ~~straightforward massage can assist to get rid of~~ the ~~discomfort and muscle cramping~~. ~~Whether you happen to be going to a massage therapist~~ or ~~just relaxing in one particular of these massaging chairs, receiving~~ a ~~massage can aid to loosen~~ the ~~muscle tissues and subsequently relieve the pain of a back ache~~.<br><br>~~In some circumstances, back discomfort sufferers find that acupuncture is a good therapy for their ailment. The jury is nevertheless out on acupuncture, and it may~~ be ~~a small as well pricy depending on how available it is in your area, but thousands of back pain sufferers swear by~~ the ~~final results of~~ the ~~needles.~~<br><br>~~Undoubtedly, one~~ of the ~~ideal attainable approaches to relieve back discomfort is to exercise often. You do not have to turn into a cardio enthusiast or a quasi weight-lifter~~, ~~but working out every single day will perform wonders in relieving back discomfort linked with cramping muscles. The physical activity can genuinely assist to get rid of the discomfort.<br>~~<br>~~If you are pregnant and suffering from back discomfort~~, ~~take into account a maternity belt to alleviate some of the discomfort~~. ~~Be taught further on this affiliated essay by visiting [http://newtonpxkf~~.~~wordpress~~.~~com My Tiny Kingdom \| Mashable]. The developing belly can really make standing straight hard, but a maternity belt supports the stomach~~, ~~thereby reducing strain on~~ the ~~back. Comfortable, low-heeled footwear are equally critical for great posture.~~<br><br>~~Try making use of coolness and heat to relieve back pain. An ice pack can~~ be ~~employed to reduce localized pain and inflammation. Warmth will function to sooth muscle tissues and market~~ the ~~flow~~ of ~~blood, ultimately assisting to heal~~ the ~~area. Dig up new information on our partner website - Click this web page: [http:~~//~~charleybjqm.wordpress.com Martini"s Persistence & A Smile~~ \| ~~Raw Replay]. Apply a heating pad or electric blanket to your back or take a warm bath ahead~~ of ~~bed. Cease the therapy prior to going to sleep.~~<br><br>As your teacher may have told you when you were young, you shouldn"t have poor posture, so work to keep the appropriate posture if you want to support ease your back pain. Usually strive to preserve your back straight, your shoulders squared, and ~~your head high. This is the body"s natural position.~~<br><br>~~If you are sitting in~~ a ~~chair working at~~ a ~~desk~~, ~~make certain your chair has very good support and is~~ the ~~right height. You ought to never ever~~ have ~~to strain to get on or off a chair. If you are not comfy, any office provide store will carry a selection~~ of ~~chairs.~~<br><br>~~Back discomfort sufferers know all too well the difficulty~~ and ~~misery their situation causes in their each day lives. However, by acquiring~~ the ~~right understanding about how~~ to ~~approach~~ the ~~dilemma, it~~ is ~~feasible to attain relief~~. ~~Take~~ the ~~guidance in this post to heart~~, and ~~you will have~~ the ~~energy~~ to ~~drastically minimize~~ the ~~impact back pain has on your life..<br><br>If you treasured this article~~ and ~~also you would like to be given more info about health reports -~~ [~~http~~:~~//slowcharlatan8074.blogs.experienceproject.com this post~~] ~~- kindly visit our webpage.~~		The [[EM algorithm]] can be treated as a special case of the MM algorithm. However, in the EM algorithm complex [[conditional expectation]] and extensive analytical skills are usually involved, while in the MM algorithm convexity and inequalities are our major focus, and it is relatively easier to understand and apply in most of the cases.

			==History==
			The original idea of the '''MM algorithm''' can be dated back at least to 1970 when Ortega and Rheinboldt were doing their studies related to [[line search]] methods.<ref>
			{{cite journal
			\|last1=Ortega \|first1=J.M.
			\|last2=Rheinboldt\|first2=W.C.
			\|title=Iterative Solutions of Nonlinear Equations in Several Variables
			\|journal=[[New York: Academic]]
			\|year=1970 \|pages=253–255
			}}
			</ref> The same idea kept reappearing under different guise in different areas since, until 2000 Hunter and Lange put all in to a general frame works and named '''MM''' for the first time.<ref>
			{{cite journal
			\|last1=Hunter\|first1=D.R.
			\|last2=Lange\|first2=K.
			\|title=Quantile Regression via an MM Algorithm
			\|journal=[[Journal of Computational
			and Graphical Statistics]]
			\|year=2000 \|volume=9 \|pages=60–77
			}}
			</ref> Recently studies have shown that it can be used in a wide range of context, like [[mathematics]], [[statistics]], [[machine learning]], [[engineering]], etc.

			==How it works==
			'''MM algorithm''' works by finding a surrogate function that minorizes or majorizes the objective function. Optimizing the surrogate functions will drive the objective function upward or downward until a local [[optimum]] is reached.

			Take the '''minorize-maximazation''' version for example.

			Let <math> f(\theta) </math> be the objective convex function we want to maximize. At the <math> m </math> step of the algorithm, <math> m=0,1... </math>, the constructed function <math> g(\theta\|\theta_m) </math> will be called the minorized version of the objective function (the surrogate function) at <math> \theta_m </math> if

			<math> g(\theta\|\theta_m)</math> ≤ <math> f(\theta) </math> for all <math> \theta </math>
			<math> g(\theta_m\|\theta_m)=f(\theta_m) </math>

			Then we maximize <math> g(\theta\|\theta_m) </math> instead of <math> f(\theta) </math>, and let

			<math> \theta_{m+1}=\arg\max_{\theta}g(\theta\|\theta_m) </math>

			The above iterative method will guarantee that <math> f(\theta_m) </math> will converge to a local optimum or a saddle point as <math> m </math> goes to infinity, because by the construction we have
			<math> f(\theta_{m+1})</math> ≥ <math> g(\theta_{m+1}\|\theta_m)</math> ≥ <math> g(\theta_m\|\theta_m)= f(\theta_m)</math>

			The marching of <math>\theta_m </math> and the surrogate functions relative to the objective function is shown on the Figure [[File:Mmalgorithm.jpg\|right\|thumb\|MM algorithm]]

			We can just flip the image upside down, and that would be the methodology while we are doing '''Majorize-Minimization'''.

			==Ways to construct surrogate functions==
			Basically, we can use any inequalities to construct the desired majorized/minorized version of the objective function, but there are several typical choices
			* [[Jensen's inequality]]
			* [[Convexity inequality]]
			* [[Cauchy–Schwarz inequality]]
			* [[Inequality of arithmetic and geometric means]]

			==References==
			{{reflist}}

			[[Category:Optimization algorithms and methods]]

70.247.162.84: link

2012-08-31T12:55:35Z

link

New page

Back pain is anything that can be a really debilitating affliction for numerous. The crucial to conquering the inconvenience and agony of this condition is to learn as significantly as attainable about its causes and therapies. Put the suggestions and info in this write-up to function, and you will stand a very good possibility of significantly increasing your high quality of life. 

Use your legs anytime you lift anything. You have to have a powerful base with your legs and your physique requirements to be even. Hold the object you are lifting up close to you as you lift from your legs. This will help avoid any back injury that may well happen. If you read for extended amounts of time every single day - either for private pleasure or specialist factors - then you can keep away from back pain resulting from neck strain by keeping your head level and bringing the documents up to that level. Keeping your head bent or raised at unnatural angles for extended amounts of time can lead to strain. So getting a document hanger or holding your materials up as an alternative of setting them on a desk or in your lap can support to avoid cumulative neck injuries associated to these posture no-no"s. Finding out to lift correctly can save oneself a lot of back pain in the future. Understanding to lift from the knees instead of just bending over to lift will tremendously reduce the strain on your back. This will maintain your back from being over exerted and assist you keep your back from establishing pain earlier. Getting back pain? Get a massage. Acquiring a back massage will ease the sore tensed muscle tissues in your back, and aid to relieve the tension of back pain. Possessing a 30 minute massage either by a professional or a loved ones member can create lengthy lasting relief from sore back muscle tissues. No matter what the explanation, if you suffer from back discomfort and you have to bend over, be sure to do so with your knees and not your back. Learn further on the affiliated essay by visiting [http://guzmanyeet.wordpress.com All Diva Media Blog | Felix Salmon - A Slice of Lime in the Soda]. A lot of men and women endure from back strains or pains due to the fact they bend over utilizing their back, which puts also much stress on the spine. Chronic muscle pain, such as back pain, can be brought on by a vitamin D deficiency. To get your fair share of this vitamin, consume lots of tiny-boned fish, fortified milk and cereal. Also, be positive to get frequent exposure to sunlight and do not forget to use that sunscreen! If you endure from chronic back discomfort, getting a straightforward massage can assist to get rid of the discomfort and muscle cramping. Whether you happen to be going to a massage therapist or just relaxing in one particular of these massaging chairs, receiving a massage can aid to loosen the muscle tissues and subsequently relieve the pain of a back ache. In some circumstances, back discomfort sufferers find that acupuncture is a good therapy for their ailment. The jury is nevertheless out on acupuncture, and it may be a small as well pricy depending on how available it is in your area, but thousands of back pain sufferers swear by the final results of the needles. Undoubtedly, one of the ideal attainable approaches to relieve back discomfort is to exercise often. You do not have to turn into a cardio enthusiast or a quasi weight-lifter, but working out every single day will perform wonders in relieving back discomfort linked with cramping muscles. The physical activity can genuinely assist to get rid of the discomfort. If you are pregnant and suffering from back discomfort, take into account a maternity belt to alleviate some of the discomfort. Be taught further on this affiliated essay by visiting [http://newtonpxkf.wordpress.com My Tiny Kingdom | Mashable]. The developing belly can really make standing straight hard, but a maternity belt supports the stomach, thereby reducing strain on the back. Comfortable, low-heeled footwear are equally critical for great posture. Try making use of coolness and heat to relieve back pain. An ice pack can be employed to reduce localized pain and inflammation. Warmth will function to sooth muscle tissues and market the flow of blood, ultimately assisting to heal the area. Dig up new information on our partner website - Click this web page: [http://charleybjqm.wordpress.com Martini"s Persistence & A Smile | Raw Replay]. Apply a heating pad or electric blanket to your back or take a warm bath ahead of bed. Cease the therapy prior to going to sleep. As your teacher may have told you when you were young, you shouldn"t have poor posture, so work to keep the appropriate posture if you want to support ease your back pain. Usually strive to preserve your back straight, your shoulders squared, and your head high. This is the body"s natural position. If you are sitting in a chair working at a desk, make certain your chair has very good support and is the right height. You ought to never ever have to strain to get on or off a chair. If you are not comfy, any office provide store will carry a selection of chairs. Back discomfort sufferers know all too well the difficulty and misery their situation causes in their each day lives. However, by acquiring the right understanding about how to approach the dilemma, it is feasible to attain relief. Take the guidance in this post to heart, and you will have the energy to drastically minimize the impact back pain has on your life.. If you treasured this article and also you would like to be given more info about health reports - [http://slowcharlatan8074.blogs.experienceproject.com this post] - kindly visit our webpage.