Competitive Learning!! Lecture 10!
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1 Competitive Learig!! Lecture 10! 1!
2 Competitive Learig! g A form of usupervised traiig where output uits are said to be i competitio for iput patters! Durig traiig, the output uit that provides the highest activatio to a give iput patter is declared the weights of the wier ad is moved closer to the iput patter, whereas the rest of the euros are left uchaged" This strategy is also called wier-take-all sice oly the wiig euro is updated" Output uits may have lateral ihibitory coectios so that a wier euro ca ihibit others by a amout proportioal to its activatio level" x 1 O1 x 2 O2 O3 x d 2!
3 Competitive Learig! g With ormalized vectors, the activatio fuctio of the i th uit ca be computed as the ier product of the uitʼs weight vector w i ad a particular iput patter x (!! Note: the ier product of two ormal vectors is the cosie of the agle betwee them" g The euro with largest activatio is the adapted to be more like the iput that caused the excitatio! " g i (x ( ) = w i T x ( w i (t +1) = w i (t) +"x ( Followig update, the weight vector is reormalized ( w =1)" 3!
4 Competitive Learig! g If weights ad iput patters are u-ormalized, the activatio fuctio becomes the Euclidea distace!! #( ) 2 g i (x ( ) = w i " x i ( g The learig rule the become! i w i (t +1) = w i (t) +"(x ( # w i (t)) 4!
5 Competitive Learig! g Competitive Learig Algorithm! 5!
6 Competitive Learig! g Demo:! 6!
7 Directio maps! 7! mcb.berkeley,edu/!
8 Tootopic maps! 8!
9 Phatom Digits! 9!
10 Kohoe Self Orgaizig Maps! g Kohoe Self-Orgaizig Maps (SOMs) produce a mappig from a multidimesioal iput space oto a lattice of clusters (or euros)! The key feature i SOMs is that the mappig is topology-preservig, i that eighborig euros respod to similar iput patters" SOMs are typically orgaized as oe- or two- dimesioal lattices (i.e., a strig or a mesh) for the purpose of visualizatio ad dimesioality reductio" g Ulike MLPs traied with the back-propagatio algorithm, SOMs have a strog eurobiological basis! O the mammalia brai, visual, auditory ad tactile iputs are mapped ito a umber of sheets (folded plaes) of cells [Gallat, 1993]" Topology is preserved i these sheets; for example, if we touch parts of the body that are close together, groups of cells will fire that are also close together" g Kohoe SOMs result from the syergy of three basic processes! Competitio" Cooperatio" Adaptatio" 10!
11 Competitio! g g Each euro i a SOM is assiged a weight vector with the same dimesioality d as the iput space! Ay give iput patter is compared to the weight vector of each euro ad the closest euro is declared the wier! The Euclidea metric is commoly used to measure distace" 11!
12 Cooperatio! g g g The activatio of the wiig euro is spread to euros i its immediate eighborhood! This allows topologically close euros to become sesitive to similar patters" The wierʼs eighborhood is determied o the lattice topology! Distace i the lattice is a fuctio of the umber of lateral coectios to the wier (as i city-block distace)" The size of the eighborhood is iitially large, but shriks over time! A iitially large eighborhood promotes a topology-preservig mappig" Smaller eighborhoods allows euros to specialize i the latter stages of traiig" 12!
13 Adaptatio! g Durig traiig, the wier euro ad its topological eighbors are adapted to make their weight vectors more similar to the iput patter that caused the activatio! The adaptatio rule is similar to the oe preseted i slide 4" Neuros that are closer to the wier will adapt more heavily tha euros that are further away" The magitude of the adaptatio is cotrolled with a learig rate, which decays over time to esure covergece of the SOM" 13!
14 SOM Algorithm! 14!
15 SOM Example(1d)! 15!
16 SOM Example(2d)! 16!
17 SOM Demo! 17!
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