Nonlinear data mapping by neural networks


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1 Nonlnear data mappng by neural networks R.P.W. Dun Delft Unversty of Technology, Netherlands Abstract A revew s gven of the use of neural networks for nonlnear mappng of hgh dmensonal data on lower dmensonal structures. Both, unsupervsed and supervsed technques are consdered. Keywords: Nonlnear data mappng, neural networks, supervsed, unsupervsed, feedforward networks, selforgansng maps, vector quantzaton Introducton to nonlnear mappng The analyss of hghdmensonal data has an ncreasng mportance due to the growth of sensor resoluton and computer memory capacty. Typcal examples are mages, speech sgnals and multsensor data. In the analyss of these sgnals they are represented n ther entrety or part by part n a sample space. As an example, a sngle data pont may represent a 6x6 (sub)mage. A collecton of these mages consttutes a cloud of ponts n a 256dmensonal space. In most multsensor data sets there s a large dependency between the sensors or between the sensor elements. Ths s certanly true for nearby mage pxels. But also more fundamentally, t s not to be expected that any physcal experment contans hundreds of degrees of freedom that are of sgnfcant mportance. Consequently, multdmensonal data sets may be represented by lower dmensonal descrptons. There are varous reasons why such representatons may be of nterest. They may reveal the structure of the data or the problem, they may be used for relatng ndvdual data ponts to each other (e.g. fndng the most smlar one n a database) or they may be used for retrevng mssng data values. Below ths wll be defned more formally. The followng representatons of a set of kdmensonal data ponts x (x, x 2,..., x j,..., x k ), x R k defned by X {x,x 2,..., x,... x m wll be dscussed here: a. A set of prototypes Z {z,z 2,..., z,... z n wth z R k,.e. a small number of data ponts n the same data space, n < m, selected or constructed n such a way that each of the orgnal ponts x s nearby one of the prototypes z. So Z s chosen such that:  mn m j { x z j () s mnmsed.the dmensonalty of data space tself s not reduced. The reducton s realsed by assgnng new data ponts x to one out of a set of known prototypes. y argmn z j { x z j (2) Ths mght be used, for example, n codng applcatons. Sets of prototypes are therefore also addressed as codebooks. b. A subspace y g(x,θ) wth y R k, θ R k, k < k,.e. a structure havng a lower dmensonalty than the orgnal data space. Each of the data ponts wll be close to or n such a subspace. A natural crteron s the mean square error:
2  x m y 2 (3) c. A probablty densty functon f(x). Ths gves for each pont n the data space the probablty densty that a data pont s found there. If ths s done well the denstes of the data ponts n X wll be hgh, as measured by the maxmum lkelhood crteron: log( f( x )) In each of these cases we have a crteron for optmzng a map M, a rule for mappng new data ponts on M and a dstance D M (x) of that pont to the map. A partcular way to apply ths s the completon of ncomplete data. Let a data vector x consst out of a known part x and an unknown part y, so x (x,y). For a gven map M one can now fnd the vector y that mnmses D M (x): (4) ŷ argmn y ( D M ( x) ) argmn y ( D M ( x', y) ) (5) It has to be realsed that the above way for optmzng M (e.g. Equaton 3) s not focused on mnmsng the error n ŷ as t uses the dstance between the map and the entre vector x. In contrast to ths unsupervsed tranng of maps the technque of supervsed tranng can be defned usng some error measure E( ŷ,y) between an estmated and a desred vector y: Choose M such that for the set of gven data vectors {(x,y ),,m E( argmn y ( D M ( x', y), y )) s mnmum. (6) 2 Neural network mappng Nonlnear maps can be represented by neural networks. These are collectons of almost dentcal smple operators (neurons) each contrbutng partally to the map, e.g.: y ( N j ( x), j, n) (7) y y 2 n whch N j (x) s a sngle neuron computng one component of y. Such a set of parallel neurons may be nested: y ( N j2 ( N j ( x), j, n ), j, n 2 ) (8) In ths case the orgnal data vectors x are mapped on a frst layer of n nput neurons. Ther outputs are mapped on a second layer of n 2 output neurons. Thereby y has n 2 Fgure : 2layer neural network components. Ths can be represented n a graphcal way lke gven n Fgure. The neurons n the frst (nput) layer have k nputs (here k 7). The neurons n the second (output) layer have 5 nputs. All neurons have just a sngle output value. So ths network maps a 7dmensonal space frst on a 5dmensonal space and then on 2dmensonal space. x x k 2
3 In each neuron a dfferent set of weghts may be stored that makes ther operaton dfferent from the other neurons. If a neuron has k nputs the sze of ths set s usually k+: one weght for each nput, consttutng a kdmensonal vector w and an addtonal constant w 0, often called bas. Two very common types of neurons are the the spatal neurons and the drectonal neurons. The spatal neurons can be thought to be located at a partcular pont n ther nput space. Ther weghts represent ths locaton. The bas may represent a sze parameter determnng the sze of some nfluence sphere. Such a neuron may thereby be defned as: N( xww, 0 ) exp{ x w w 0 (9) Ths neuron yelds a hgh output value for nput data vectors that are close to ther locaton (relatve to w 0 ). Ths neuron transfer functon s called a radal bass functon. The drectonal neurons can be thought to represent a drecton n ther nput space. Ther outputs are related to the angles between the nput data vectors and ther weght vectors, e.g.: N( xww, 0 ) exp{ w x w 0 (0) Ths neuron transfer functon s called a sgmodal functon or just sgmod. Note that the transfer functons as defned by Equaton 9 and Equaton 0 yeld values between 0 and. It s very common the scale the values wthn neural network layers n ths way. Sometmes they are scaled between  and +. In order to be able to produce arbtrary output values n the output layer lnear transfer functon have to be used: N ( xww, 0 ) w x+ w 0 () The feedforward network as gven n Fgure can be used for all three types of mappngs gven n the frst secton. Prototypes can be represented by radal bass functons n the nput layer. The dstance between an nput vector and the set of prototypes s defned n the output layer. If sgmodal outputs are used the network represents some probablty densty functon f the outputs are correctly normalsed. A feedforward networks wth just sgmodal transfer functon represents a mappng by nonlnear subspaces. Another common type of neural networks s the selforgansng map (SOM) or Kohonen network as shown n Fgure 2. In ths network of spatal neurons, the neurons are not connected. They thereby represent a set of prototypes. Durng the optmzaton of the network some connectvty s assumed. Ths wll further be dscussed below. Fgure 2: Archtecture of a 2D selforgansng map. 3 Unsupervsed technques We wll now gve some examples on how neural networks can be traned n an unsupervsed way. The three mappng types gven n the ntroducton wll be treated separately. 3
4 3. Prototypes by vector quantzaton Suppose we have an ntal set of prototypes (e.g. a random selecton of the orgnal data set) Z{z,z 2,..., z,... z n. A very smple way of updatng ths set s by determnng the subsets X X,,n such that X contans all vectors n X that have z as ther nearest prototype. Now all prototypes are replaced by the means of the data vectors n the correspondng X. Ths s reterated untl stablty. Ths procedure s called the kmeans method. A sequental procedure can be defned by: z j z j + α(t)[x  z j ] f z j s the nearest prototype of x (2) repeat for,m teratvely repeat wth decreasng α(t). Formally ths procedure has not much relaton wth neural networks. It s, however, for hstorcal reasons treated n connecton wth these. 3.2 Subspaces by neural networks 3.2. Subspaces by selforgansng maps (SOMs) Subspaces can be found n relaton wth the above treated vector quantzaton method by condtonng the set of prototypes n a grd of the desred subspace dmensonalty. Most popular s the use of a n x n grd of n 2 neurons representng n 2 prototypes. Ths structure s emphaszed and preserved by replacng the above update rule by: z j z j + α jk (t)[x  z j ] f z j s the nearest prototype of x and z k s wthn some predefned grd neghbourhood of z j. α jk (t) s a weghtng functon that decreases wth the number of teratons and wth the grd dstance between z j and z k. Ths procedure may be followed for structures of any dmenson. In practce, however, just one, two, and sometmes threedmensonal grds are used. For more dmensons too many neurons wll be needed Subspaces by dabolo feedforward networks n 4 k n 3 n n 2 n Feedforward networks as gven n Fgure represent n each layer a subspace. If t s possble to map the nput data on a partcular layer a reconstructon by followng layers x should be possble, see Fgure 3. The 3rd and the 4th layer x k Fgure 3: 4layer dabolo network restore the orgnal data vectors after they are mapped on the 2nd layer. Tranng should be done by mnmzng the dfference between nputs and outputs of the entre network. In fact ths s a supervsed procedure, see below. The data set tself, however, does not contan any target values. Thereby ths s a supervsed soluton of a nonsupervsed problem. 4
5 3.3 Probablty densty functons In ths case the neural network outputs are consdered as probablty denstes. For that reason the output layer consst of a sngle neuron and has a lnear transfer functon. Usually the frst layer has radal bass transfer functons. A good network yelds hgh outputs for the tranng data vectors. The maxmum lkelhood crteron s thereby often chosen for optmzng the network: log( ) y (3) 4 Supervsed technques We wll now gve some examples on how neural networks can be traned n an supervsed way. Recall that n ths case the gven data set X conssts of vectors x (x,y) and that networks have to be found that estmate y as correctly as possble f the part x s gven. 4. Prototypes by learnng vector quantzaton The qualty of prototypes has to be judged on the bass of the dstance between the desred and the estmated yvalues (targets). Equaton 2 mght therefore be replaced by somethng lke z j z j + Α(t)[x  z j ] f z j s the nearest prototype of x (4) n whch the matrx A(t) has larger values for the target components. 4.2 Subspaces by neural networks Ths problem s the tradtonal use of the feedforward network, Fgure. The network outputs are the target values. Agan lnear output transfer functons mght be needed. A dabolo network s not necessary now. A wde range of network optmzaton technques s avalable. Tradtonally backpropagaton s used, an teratve gradent descent technque. Faster technques, especally for smaller networks, mght be preferred. 4.3 Probablty densty functons The unsupervsed soluton can be appled also here. If the jont probablty densty functon for x [x,y] s estmated by f(x,y), the yvalue for a gven x mght be found by ŷ argm ax y ( f( x', y) ) (5) References.C. Mao and A.K. an, Artfcal neural networks for feature extracton and multvarate data projecton, IEEE Transactons on Neural Networks, vol. 6, no. 2, 995, A.K. an,. Mao, and K.M. Mohuddn, Artfcal Neural Networks: A Tutoral, Computer, vol. 29, no. 3, pp. 344, T. Kohonen, Selforganzng maps, Sprnger, Berln, C.M. Bshop, Neural networks for pattern recognton, Clarendon Press, Oxford, B.D. Rpley, Pattern recognton and neural networks, Cambrdge Unversty Press, Cambrdge,
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