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Topic: Local minima

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Maxima and minima

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	PlanetMath: extremum
		If you imagine the function as a surface, then a local minimum is the bottom of a “valley” or “bowl” in the surface somewhere.
		A “strict local minima” or “strict local maxima” means that nearby points are strictly less than or strictly greater than the critical point, rather than
		Finding minima or maxima is an important task which is part of the field of optimization.
	planetmath.org /encyclopedia/LocalMaxima.html (222 words)

	Identifying image structures for content-based retrieval of digitized spine x-rays
		Localization corresponds to the understanding of the image objects at a coarser level than segmentation, where object boundaries are determined in detail.
		Two technical difficulties remain: (a) this set of local minima is may include local minima that are not between the spinous processes at all, but are actually in the skull region.
		This sequential localization and identification of structures is based on location of previously-determined structures, and a prior statistical data on the relative geometries of the vertebral structures.
	archive.nlm.nih.gov /pubs/long/spie-sd2002/spie-sd2002.php (4034 words)

	Minimum / Minima, and other Irregular Plurals
		Thus we can talk about x = a being a local minimum of a function f(x), meaning that for x = a, f(x) has a value that is lower than for nearby values of x.
		In particular, it doesn't make sense to talk about ??"a global minima" (because "a" implies singular, and "minima" implies plural) or to talk about "global minima" unless you are talking about minima of two or more different functions.
		"Local minima" of a function makes sense (assuming the function has more than one local minimum), but ??"a local minima" and ??"a minima" still are not right, as once again, "a" implies singular, and "minima" implies plural.
	www.cse.unsw.edu.au /~billw/cs9444/minima.html (281 words)

	Maxima and minima - Wikipedia, the free encyclopedia
		In mathematics, maxima and minima, also known as extrema, are points in the domain of a function at which the function takes the largest (maximum), or smallest (minimum) value either within a given neighbourhood (local extrema), or on the function domain in its entirety (global extrema).
		In order to be able to define local maxima and local minima, the function needs to take real values, and the concept of neighborhood must be defined on the domain of the function.
		One refers to a local maximum/minimum as to a local extremum (or local optimum), and to a global maximum/minimum as to a global extremum (or global optimum).
	en.wikipedia.org /wiki/Maxima_and_minima (681 words)

	DIGITAL VIDEO: COMPRESSION: BMAs: local minima (Site not responding. Last check: 2007-10-19)
		A local minimum is a candidate block location where the neighbouring blocks have greater distortion but the candidate block is not the best of the entire search area.
		The quadrant monotonic assumption and the problem of local minima are analogous to a mountain climber making the assumption that from an arbitrary starting position he will eventually arrive at the highest point as long as he continues to climb up (Illustrated above).
		Local minima are common in images and the local minima are typically not as good as the global minimum.
	www.newmediarepublic.com /dvideo/compression/adv11.html (296 words)

	[No title]
		Hence for local search algorithms, such as gradient descent, the quality (optimality) of the final solution is highly dependent upon the selection of the initial search point.
		The function has three minima in the region x [0.05, 0.5], as is shown in Figure 8.1.1, which are located approximately at 0.058, 0.091, and 0.223.
		Also, stochastic gradient search has a better chance of escaping local minima and areas of shallow gradient, which may allow it to converge faster (Hoptroff and Hall, 1989).
	neuron.eng.wayne.edu /tarek/MITbook/chap8/8_1.html (2081 words)

	Removing Transversality Assumption (Site not responding. Last check: 2007-10-19)
		A GVG edge is now any point with at least m closest equidistant local minima such that the minima only reside in an m - 1 dimensional set, while a GVG meetpoint is defined as any point with at least m + 1 local minima such that the minima reside in an m dimensional set.
		Additionally, the local minima reside on a sphere of the meetpoint is a center.
		Each face f on the convex hull has a normal n and a local minima mf that form f; this normal is a direction which differentially maintains equidistance to mf.
	voronoi.sbp.ri.cmu.edu /~dbender/hgvg/transversal.html (881 words)

	SparkNotes: Applications of the Derivative (BC): Analysis of Graphs
		Therefore, in order to find the local minima/maxima of a function, we simply have to find all its critical points and then check each one to see whether it is a local minimum, a local maximum, or neither.
		There may be critical points of a function that are neither local maxima or minima, where the derivative attains the value zero without crossing from positive to negative.
		Once we have found the critical points, one way to determine if they are local minima or maxima is to apply the first derivative test.
	www.sparknotes.com /math/calcbc1/applicationsofthederivative/section2.rhtml (898 words)

	[No title]
		Therefore, the maxima is.577, and the minima is -.577.
		So the local maxima and minima in the critical cases are the positive and negative square roots of 1/3.
		the local maximum is at -square root of (1/3) and the local minimum is at +square root of (1/3).
	www.mtholyoke.edu /~mpeterso/classes/math101/jitt15ans.html (1083 words)

	Global Optimization: Package Functions
		It is designed to be robust to local minima and to solve problems with hundreds of variables.
		It is designed to be robust to local minima, and the problems can have inequality constraints.
		It is designed to be robust to local minima and to solve midsized problems (up to 15 variables).
	www.wolfram.com /products/applications/globalopt/functions.html (595 words)

	[No title] (Site not responding. Last check: 2007-10-19)
		Local "valleys" or minor "dents" in a loss function which, in many practical applications, will produce extremely large or small parameter estimates with very large standard errors.
		The Simplex method is particularly effective in avoiding such minima; therefore, this method may be particularly well suited in order to find appropriate start values for complex functions.
		Robust locally weighted regression is a method of smoothing 2D scatterplot data (pairs of x-y data).
	www.statsoft.com /textbook/glosl.html (4798 words)

	[No title]
		Intuitively, the existence of local minima is due to the fact that the error function E is the superposition of functions that may have minima at different points.
		In order to actually exhibit such an example, it is necessary to obtain terms whose minima are far appart, and to control the second derivatives in such a way that the effect of such minima is not cancelled by the other terms (as happens in the case of convex functions).
		Properties of local minima of rational functions are decidable (theory of real-closed fields), so in principle this change of variables is of some interest besides the role that it plays in the present paper.
	www.math.rutgers.edu /~sontag/complex_systems.html (4184 words)

	Relating Tuning and Timbre
		Property 3: The dissonance curve generated by F has at most 2n(n-1) local minima which are located symmetrically (on a logarithmic scale) so that half occur for intervals between 0 and 1, and half occur for intervals between 1 and infinity.
		In figures 4(b) and 4(c), for instance, local minima are found at a=1.15 and a=1.86 respectively, which is the ratio between the two partials.
		The principle of local consonance says that the most appropriate scale tones for harmonic timbres are located at such a, and indeed, all the points of local consonance of figure 3 occur at such values.
	eceserv0.ece.wisc.edu /~sethares/consemi.html (4740 words)

	Separatrices
		It allows to define the set of minima (maxima for watercourses) to be considered instead of the minima of the analyzed image, which reduce oversegmentation and drive the algorithm to interesting structures.
		A suitable set of minima consist of the center of the flesh ring (ridge-like SA), the center of the bone (valley-like SA), any point inside the brain and the border of the image (figure 8).
		For example, in the medialness measure of figure 5(b) there are not local minima and, therefore, the watershed algorithm, for example, can not extract the ridge-like structure.
	homepages.inf.ed.ac.uk /rbf/CVonline/LOCAL_COPIES/LOPEZ/node6.html (2117 words)

	Find local maxima and minima (Site not responding. Last check: 2007-10-19)
		When the graph of the function f, continuous at x=c, is increasing on the immediate left of the number x=c and decreasing on the immediate right of the number x=c, then the value of f at c is locally the largest, i.e., f(c) is a local maximum.
		When the graph of the function f, continuous at x=c, is decreasing on the immediate left of the number x=c and increasing on the immediate right of the number x=c, then the value of f at c is locally the smallest, i.e., f(c) is a local minimum.
		For continuous functions, the local maxima and local minima can only occur at the critical numbers or endpoints of the domain of f.
	deadline.3x.ro /maxima-minima.html (529 words)

	R: Some diagnostics for a fitted gam model
		Although not conclusive (except in the single smoothing parameter case), a lack of multiple local minima on this plot is suggestive of a lack of multiple local minima in the GCV/UBRE function and is therefore a good thing.
		Multiple local minima on this plot indicates that the GCV/UBRE function may have multiple local minima, but in a multiple smoothing parameter case this is not conclusive - multiple local minima on one slice through a function do not necessarily imply that the function has multiple local minima.
		These minima usually imply a big change in model parameters, and have the characteristic that the minimia will not be present in the GCV/UBRE score of the approximating model that would result from actually applying this parameter change.
	math.furman.edu /~dcs/courses/math47/R/library/mgcv/html/gam.check.html (768 words)

	A function with a single critical point, which is a local minimum but not a global minimum (Site not responding. Last check: 2007-10-19)
		A function with a single critical point, which is a local minimum but not a global minimum
		For a function of a single variable f(x), if f is continuous on an interval I, has only one critical point in I, and that critical point is a local minimum, then it is the absolute (or global) minimum.
		The only point solving both of these equations is (0,1), and the second derivative test shows that this point is a local minimum for f.
	www.math.tamu.edu /~tom.vogel/gallery/node16.html (214 words)

	When Seysen's Algorithm Fails
		As the number of local minima increases, it is increasingly likely that in the process of reducing lattice L the S(A) function (where A is the quadratic form associated with L) will encounter one of these local minima.
		For many types of lattices, such as the sparse lattices generated by subset sum problems (see Chapter 3), Seysen's algorithm has performed sufficient work by the time it encounters a local minimum that it is acceptable for it to stop.
		If one considers the surface described by S(A) values, local minima are ``wells'' or ``depressions'' in the surface which are large enough to contain all points reachable by performing one row move on the lattice.
	www.farcaster.com /papers/sm-thesis/node19.html (327 words)

	Local or Global Minima: Flexible Dual-Front Active Contours (Site not responding. Last check: 2007-10-19)
		Most variational active contour models are designed to find local minima of data-dependent energy functionals with the hope that reasonable initial placement of the active contour will drive it towards a ``desirable'' local minimum as opposed to an undesirable configuration due to noise or complex image structure.
		As such, there has been much research into the design of complex region-based energy functionals that are less likely to yield undesirable local minima when compared to simpler edge-based energy functionals whose sensitivity to noise and texture is significantly worse.
		By simply adjusting the size of active regions, the ability to gracefully move from capturing minima that are more local (according to the initial placement of the active contour/surface) to minima that are more global makes this model much easier to obtain "desirable" minimizers (which often are neither the most local nor the most global).
	users.ece.gatech.edu /~huali/dfm.html (338 words)

	SparkNotes: Applications of the Derivative (BC): Analysis of Graphs
		Therefore, in order to find the local minima/maxima of a function, we simply have to find all its critical points and then check each one to see whether it is a local minimum, a local maximum, or neither.
		There may be critical points of a function that are neither local maxima or minima, where the derivative attains the value zero without crossing from positive to negative.
		Once we have found the critical points, one way to determine if they are local minima or maxima is to apply the first derivative test.
	sparknotes.com /math/calcbc1/applicationsofthederivative/section2.rhtml (900 words)

	OptiVec: VF_localminima (Site not responding. Last check: 2007-10-19)
		The indices of local minima in X are stored in Ind and the number of local minima is returned (this is the number of elements of Ind).
		A local minimum is defined as one element of X that is smaller than both its neighbours to the right and to the left.
		That means that the zero'th and the last element of X (which have only one neighbour) cannot be local minima.
	www.optivec.com /vecfuncs/localminima.htm (86 words)

	Journal of Vision - A gradient based heuristic for the perception of 3D shape from texture, by Todd & Thaler
		maxima or minima) in any given direction will be optically specified by local minima in the spatial frequency of the projected image texture in that direction.
		This correspondence between local depth extrema on a surface and local minima of spatial frequency in an image does not necessarily occur, however, for surfaces with anisotropic textures.
		The results revealed that that the perceived locations of local depth extrema were more highly correlated with the positions of local spatial frequency minima in the projected image texture, than with the locations of the actual depth extrema on the depicted surface.
	www.journalofvision.org /5/8/995 (360 words)

	Local Minima (Site not responding. Last check: 2007-10-19)
		The last problem which must be considered is that we can never reach the local minimum.
		Often it has been said that refinement was continued until convergence at the local minimum.
		This means that no one has ever been ``trapped in a local minimum''.
	www.uoxray.uoregon.edu /dale/papers/CCP4_1994/node16.html (87 words)

	Publications List with Abstracts - Dr Len Hamey
		The analysis proves the absence of local minima, eliciting significant aspects of the structure of the error surface.
		The present work is important for the study of the existence of local minima in feedforward neural networks, and also for the development of training algorithms which avoid or escape entrapment in local minima.
		Hamey, "Results on weight configurations that are not local minima in feed-forward neural networks," in Proceedings of the Seventh Australian Conference on Neural Networks (P. Bartlett, A. Burkitt, and R. Williamson, eds.), (Canberra, Australia), pp.
	www.ics.mq.edu.au /~len/abstracts.html (6200 words)

	[No title] (Site not responding. Last check: 2007-10-19)
		These functions both * calculate all local minima at each given point and match those local minima * to previously located local minima at the previous location (before a small * step).
		Then, by marking the minimas you want to track with ids (the ids must only be lower than the total number of minimas), you can take the next step using the update constructor, which takes in the old minima data.
		You can then give it the ids you set to get back the index of each tracked object -- if a minima that you were tracking split or joined, then it is signified by a special id that can be used to determine which objects split or joined.
	voronoi.sbp.ri.cmu.edu /software/hgvg/hgvg3d01/minima.h (821 words)

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