(*^ ::[ Information = "This is a Mathematica Notebook file. It contains ASCII text, and can be transferred by email, ftp, or other text-file transfer utility. It should be read or edited using a copy of Mathematica or MathReader. If you received this as email, use your mail application or copy/paste to save everything from the line containing (*^ down to the line containing ^*) into a plain text file. On some systems you may have to give the file a name ending with ".ma" to allow Mathematica to recognize it as a Notebook. The line below identifies what version of Mathematica created this file, but it can be opened using any other version as well."; FrontEndVersion = "X Window System Mathematica Notebook Front End Version 2.2"; X11StandardFontEncoding; fontset = title, inactive, noPageBreakBelow, noPageBreakInGroup, nohscroll, preserveAspect, groupLikeTitle, center, M7, bold, e8, 24, fontName, "times"; fontset = subtitle, inactive, noPageBreakBelow, noPageBreakInGroup, nohscroll, preserveAspect, groupLikeTitle, center, M7, bold, e6, 18, fontName, "times"; fontset = subsubtitle, inactive, noPageBreakBelow, noPageBreakInGroup, nohscroll, preserveAspect, groupLikeTitle, center, M7, italic, e6, 14, fontName, "times"; fontset = section, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeSection, grayBox, M22, bold, a20, 18, fontName, "times"; fontset = subsection, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeSection, blackBox, M19, bold, a15, 14, fontName, "times"; fontset = subsubsection, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeSection, whiteBox, M18, bold, a12, 12, fontName, "times"; fontset = text, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 12, fontName, "times"; fontset = smalltext, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 10, fontName, "times"; fontset = input, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeInput, M42, N23, bold, 12, fontName, "courier"; fontset = output, output, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, L-5, 12, fontName, "courier"; fontset = message, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, 12, fontName, "courier"; fontset = print, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, 12, fontName, "courier"; fontset = info, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, 12, fontName, "courier"; fontset = postscript, PostScript, formatAsPostScript, output, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeGraphics, M7, l34, w282, h287, 12, fontName, "courier"; fontset = name, inactive, noPageBreakInGroup, nohscroll, preserveAspect, M7, italic, B65535, 10, fontName, "times"; fontset = header, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, italic, 12, fontName, "times"; fontset = leftheader, 12, fontName, "times"; fontset = footer, inactive, nohscroll, noKeepOnOnePage, preserveAspect, center, M7, italic, 12, fontName, "times"; fontset = leftfooter, 12, fontName, "times"; fontset = help, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 12, fontName, "times"; fontset = clipboard, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 12, fontName, "times"; fontset = completions, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 12, fontName, "courier"; fontset = special1, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 12, fontName, "times"; fontset = special2, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 12, fontName, "times"; fontset = special3, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 12, fontName, "times"; fontset = special4, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 12, fontName, "times"; fontset = special5, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 12, fontName, "times";paletteColors = 128; currentKernel; ] :[font = title; inactive; preserveAspect] A Self Organizing Network for Similarity Extraction and Scaled Parametric Output (Multi-Dimensional Scaling) :[font = subsubtitle; inactive; preserveAspect; startGroup] From a term paper by John Hershey, 1991 :[font = subsection; inactive; preserveAspect; startGroup] Abstract :[font = text; inactive; preserveAspect; endGroup] A general parallel architecture and learning rule is developed to perform a self-organized mapping from higher dimensional input space to lower dimensional output space, or what is sometimes known as multidimensional scaling (MDS). The structure of the mapping is derived entirely from the inputs. The network is fast, flexible and requires few nodes compared with a topological self organizing network. :[font = subsection; inactive; preserveAspect; startGroup] Introduction :[font = text; inactive; preserveAspect] This form of network was inspired by some problems I had with Kohonen 's (1982) self organizing topological maps. I wanted a network which would adapt to the structure of incoming data in a meaningful way much like the topological map. However, Kohonen's topological maps are predisposed to certain structures by their neighborhood functions. The map can only mould itself to the input distribution if that distribution conforms to the restrictions of the map's shape. For example, a two dimensional Kohonen map would have difficulty mapping an input distribution in the form of a ring without excessive distortion and misrepresentation . That is, units in the center of the map would either gravitate toward the input ring, causing distortion and misrepresenting the distances among them, or would remain in the center, where they do not detect any meaningful feature of the input distribution. Another problem is that a large number of cells are required to map a complex shape because of the localist nature of the Kohonen map (meaning is defined by a one to one mapping of the input location to the cell's location in the network). For instance, an input distribution in the form of a helix would require a three dimensional lattice of cells, requiring a thousand cells for a resolution of ten in each dimension. If the output represented location as a mapping into activity space rather than on the basis of cell location, these problems would be eliminated. Thus, my goal was a network with the following properties: :[font = text; inactive; preserveAspect] no predefined ordering or neighborhood to the units, :[font = text; inactive; preserveAspect] all structure derived from input similarities, :[font = text; inactive; preserveAspect] a mapping into a low dimensional activity space, :[font = text; inactive; preserveAspect; endGroup] the number of required nodes related to the input distribution rather than the dimensionality of the output. :[font = subsection; inactive; preserveAspect; startGroup] General Concept :[font = text; inactive; preserveAspect; endGroup] The network I devised uses the differences of its responses to inputs as the neighborhood function rather than the physical distances of nodes used by the Kohonen architecture. The ordering and meaning of the nodes is produced by an output layer which adapts to the input nodes such that each input node produces a unique output state and that differences of output states correspond systematically to differences among the input states. This mapping stage essentially performs a multidimensional scaling (MDS) of the differences produced by the input stage. The beauty of this process is that the difference function can be whatever you like as long as it is easily computable by the network, is meaningful, and conforms to the four axioms which define a metric (Baird & Noma, 1978): 1. d (a,a) = 0 2. d (a,b) >= 0 3. d (a,b) = d (b,a) 4. d (a,b) + d (b,c) >= d (a,c) (Triangle inequality) The distance function is related to the activation function of the input node, and computes the distance between the weights and the input. Since we usually want our nodes to fire in response to their target input, we can have the nodes compute a similarity function, and then convert this to a distance function by inversion or negation. The nodes begin with random weights and essentially tune in to the nearest input. Thus the nodes remain physically unordered, but as a whole form a skeleton representation of the input distribution. In this configuration, a single input causes a pattern of excitation equivalent to the similarity functions of each unit to the input. Since there will be a winning unit with a maximized response, we can interpret the excitation pattern as proximities in weight space of each unit to that winning unit. For the output units we would like a set of weights and an activation function such that the activity space of the output units represents a low dimensional mapping of the input proximities. One way of acheiving this is to produce the scaled mapping in the output weight space, such that the weights from a single input unit to the output layer represent its location in the output map. Then, if we force only one input unit (the winner) to fire at a time (with a value of 1.0) through to the output layer, we can use the familiar linear summation unit to convert the weight space into an identical activation space. In order to acheive this, the learning algorithm compares the proximities of the node's input weights with the proximities of the output weights, and adjusts each node's output weights in order to minimize this difference. To compute the differences among the output weights it is convenient to use the same distance function as the input activation function, with the winner's output weights/activation as input and each set of output weights as a set of input weights, although this is not necessary. A useful analogy to this process is to conceive of the distances among the output weights as springs which resist both compression and expansion. The weights begin in random positions, considerably stressed. The equilibrium point of an output spring is reached when its length matches the corresponding input distance. Since the network operates on sequentially presented inputs (and possibly a dynamic distribution), and since the distances are computed by the activation function with reference to a single input/winner node, we have limited distance information and cannot simply release all nodes at once (although this would be a good approach if we knew all of the distances at once). One possible learning algorithm would be to simply release the winner node on each iteration and allow its known tensions to pull it toward the correct position. This process fails for a number of reasons. First, it is very slow to go one at a time. Second, there may be a surplus of nodes which never become winners, are never updated from their random positions, and thus fix the network in some convoluted configuration. Third, it is succeptible to local minima ( i. e. two similar nodes that are trapped on opposite sides of a wall of distant nodes may remain A much faster, but less precise way to use the same tension information is to hold the winner stationary and to release all the other units, allowing them to relax toward the correct distance from the winner, and ignoring their unknown distances from each other. This method converges very quickly if there is no tension in the final configuration. If there is tension, then the configuration will oscillate around some minimum stress configuration, since in the optimal configuration decreasing the tension to a given winner node inherently increases the tensions among the other nodes. However, as we reduce the learning rate the process approaches the dynamics of releasing all of the nodes at once, which would eliminate the oscillation. (Note-- don't take the analogy to springs too rigidly as we are not implementing acceleration, which causes real springs to oscillate. Think of springs submerged in a viscous goop.) This form of simulated annealing is necessary if you are interested in a high-stress solution; however, such high stress is inversely related to goodness of fit, so the real solution may be to increase the dimensionality of the output (Davidson 1983). :[font = subsection; inactive; preserveAspect; startGroup] I/O Representation, Activation Function, Inputs :[font = text; inactive; preserveAspect] The distance function determines the type of spatial representation used for input and output. Two somewhat conflicting demands on the distance function are ease of visual representation and neural computability. The standard linear dot-product units are fine in the neural computability department, however they are restricted to computing a correlation function (when normalized), which is a directional vector and is not Euclidean. Beyond a two dimensional spherical surface, this representation is difficult to visualize, since visualization demands projection into a higher dimensional Euclidean space. The Euclidean distance function is the most convenient to use in the present model because of the handy Euclidean structure of the interface, but it requires units with a Pythagorean activation function. The pros outweigh the cons. The activation function is thus defined in terms of the pythagorean distance of the input vector to the weights (square root of the sums of squared differences in each dimension) . The distance is transformed so that activity represents proximity, and so that the unit is very selective to a given location. The transformation is the inverse exponential of the distance squared : :[font = output; output; inactive; preserveAspect] e^(-d^2) ;[o] 2 -d e :[font = text; inactive; preserveAspect] Distance can be scaled by dividing by a scaling factor , s, which changes the spread of sensitivity of the unit. (See graph below) :[font = subsubsection; inactive; preserveAspect; startGroup] Activation and Distance Functions and Graph :[font = text; inactive; preserveAspect] This Euclidean distance function and corresponding activation function can be replaced with non-Euclidean functions, as long as the inputs and output graphics are changed to a suitable representation. The unpdate procedure may also need to be changed to work with other activation functions. :[font = input; preserveAspect] Clear[activate,distance]; s = 1; distance[v1_,v2_] := N[Sqrt[ Plus@@ ((v1 - v2)^2)]]; activate[inputV_,weightV_] := N[( d = distance [inputV,weightV]; E^-(d/s)^2 ) ]; :[font = text; inactive; preserveAspect] Here is the activation of a node with weights of {0,0} as a function of 2-D input space. :[font = input; preserveAspect; startGroup] Plot3D[activate[{x,y},{0,0}],{x,-2,2},{y,-2,2},PlotRange->{0,1}]; :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 0; pictureTop = 8; pictureWidth = 362; pictureHeight = 297; endGroup; endGroup] %! %%Creator: Mathematica %%AspectRatio: 0.82055 MathPictureStart /Courier findfont 10 scalefont setfont % Scaling calculations 0.0249355 0.99742 -0.0396341 0.99742 [ [(-2)] 0.05113 0.25884 1 0.933946 Msboxa [(-1)] 0.1926 0.20316 0.96648 1 Msboxa [(0)] 0.34275 0.1441 0.862234 1 Msboxa 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gsave 0.001 setlinewidth 0.03389 0.43363 moveto 0.03741 0.43237 lineto stroke grestore gsave 0.001 setlinewidth 0.03094 0.45224 moveto 0.03447 0.45099 lineto stroke grestore gsave 0.001 setlinewidth 0.02945 0.46163 moveto 0.03298 0.46039 lineto stroke grestore gsave 0.001 setlinewidth 0.02796 0.47108 moveto 0.03149 0.46984 lineto stroke grestore gsave 0.001 setlinewidth 0.02645 0.48058 moveto 0.02998 0.47935 lineto stroke grestore grestore % End of Graphics MathPictureEnd :[font = subsubsection; inactive; preserveAspect; startGroup] Initialization Constants & Graphics Defaults :[font = text; inactive; preserveAspect] In this example a three-dimensional input space will be mapped onto a two-dimensional map space. The algorithm is entirely general, however, and other dimensionalities can be used by changing the setup parameters and graphical displays and creating an n-dimensional set of input points. :[font = text; inactive; preserveAspect] -A note on variable names: variables suffixes denote the following: "N" means an integer number of items. "V" means a vector. "M" means a matrix. :[font = input; preserveAspect] inDimN = 3; (* input dimensions *) mapDimN = 2; (* mapping dimensions *) inNodeN = 30; (* input nodes *) inPatN = 30; (* input patterns *) rnd := N[2 Random[]-1]; (* generate random number {-1,1} *) $MessagePrePrint= Format; (* long error messages *) :[font = input; preserveAspect; endGroup] SetOptions[Graphics,AxesOrigin->{0,0}, Axes->True, AspectRatio->1, PlotRange->{{-1,1},{-1,1}}]; SetOptions[Graphics3D, Axes->True, AspectRatio->1, PlotRange->{{-1,1},{-1,1},{-1,1}}]; :[font = subsubsection; inactive; preserveAspect; startGroup] Input Function - Conical Helix Graph :[font = text; inactive; preserveAspect] This conical helix input was chosen to be squashable onto two dimensions without unacceptable stress. Note that the lines are for visualization only and are irrelevent to the inputs, and we could just as well draw radial lines on this plot. The matrix "inputM" contains an ordered list of points in input space, which will be presented at random to the network. :[font = input; preserveAspect; startGroup] inputM = Table[{ .9 Sin[6 Pi (t/inPatN) ] (1- .7 t/inPatN), .9 Cos[6 Pi (t/inPatN) ] (1- .7 t/inPatN), .6 (2 t/inPatN - .6 )}//N, {t,inPatN}]; :[font = message; inactive; preserveAspect; endGroup] General::spell1: Possible spelling error: new symbol name "inputM" is similar to existing symbol "inputV". :[font = input; preserveAspect; startGroup] Show[inPlot = Graphics3D[{ PointSize[0.02], Line[inputM], Map[Point,inputM]} ]]; :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 3; pictureWidth = 254; pictureHeight = 254; endGroup; endGroup; endGroup] %! %%Creator: Mathematica %%AspectRatio: 1 MathPictureStart /Courier findfont 10 scalefont setfont % Scaling calculations -0.0260524 1.10184 0.0254869 1.01948 [ [(-1)] 0.10149 0.23107 0.980657 1 Msboxa [(-0.5)] 0.22831 0.18149 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0.03646 0.69799 lineto stroke grestore grestore gsave grestore % End of Graphics MathPictureEnd :[font = subsection; inactive; preserveAspect; startGroup] Main Algorithm :[font = text; inactive; preserveAspect] The algorithm consists of two coevolving learning processes. The learning in the input layer is dependent only upon the inputs, while the mapping depends upon the input layer. All weights are randomized in the beginning. The input layer learns by moving nodes toward an input in proportion to their activations. Nodes contain a learning threshold such that it can only move if its activation function exceeds its threshold. Thresholds begin at zero and are moved up as the node approaches an input so that it won't be pulled away too easily (otherwise all nodes tend to be pulled toward the center of gravity of the inputs). The winning node is the node nearest the input with the greatest activation above threshold. In order to simplify and quicken the model I merely update the winning node to the input value and bump up the threshold to make it stick. For more complex or noisy data it would be better to gradually update the input nodes, for an averaging effect, but in this case its slow and unnecessary. The map layer learns by moving its mapping of each node toward or away from the winning node in proportion to the difference between the input distances and the map distances. Input distances are computed from the activation of the input layer. Map distances are computed by treating the output of the map layer as an input and calculating an activation function for each input node according to the map weights. That is, it is as if we have another layer of units, one for each input unit, computing the same type of activation function, but instead of comparing the input to the input weights, they compare the output to the output weights for each node. The difference between these two activations is used to modify the distances among the map weights such that they correspond to differences among the input weights (but in a lower dimension). For each unit, I take its map weights and subtract the winners map weights, producing the directional vector leading toward the winner. I then scale this vector by the distance to the winner in input weight space minus the distance to the winner in map weight space. This vector is then scaled by the learning rate and added to the former map weights. Each iteration of the main loop consists of: :[font = text; inactive; preserveAspect] supplying an input point, :[font = text; inactive; preserveAspect] calculating the activation of the input layer, :[font = text; inactive; preserveAspect] finding the winner, :[font = text; inactive; preserveAspect] adjusting the input weights and learning threshold, :[font = text; inactive; preserveAspect] comparing the input distances to the map distances, :[font = text; inactive; preserveAspect] adjusting the map weights :[font = subsubsection; inactive; preserveAspect; startGroup] Display Procedure showGraphics[] :[font = input; preserveAspect; endGroup] showGraphics[] := Show[GraphicsArray[{ (* Input Space Display *) Graphics3D[{ (* Show location of Input as grey disk *) RGBColor[.6,.6,.6], PointSize[0.08], Point[inpoint], (* Show skeleton of Input as helix curve *) RGBColor[1,0,0], Line[inputM], (* Show location of nodes as black dots *) RGBColor[0,0,0], PointSize[0.02], Map[Point,inputWtsM]}, PlotLabel -> "Input Space"] , (* Output Map Display *) Graphics[{ (* Show location of Winner as grey disk *) RGBColor[.6,.6,.6], PointSize[0.08], Point[mapWtsM[[winner]]], (* Show location of outputs as black dots *) PointSize[0.02], RGBColor[0,0,0], Map[Point,mapWtsM], (* Show outputs of winner nodes to ordered inputs as curve *) Line[ mapWtsM [[windexV]]]}, PlotLabel -> "Output Map"] }, PlotLabel-> "After "<>ToString[i]<>" Iterations" ] ]; :[font = subsubsection; inactive; preserveAspect; startGroup] Randomize Weights & Set Parameters :[font = input; preserveAspect] SeedRandom[1]; inputWtsM = Table [rnd, {inNodeN} ,{inDimN}]; (* input weight matrix *) mapWtsM = Table [rnd,{inNodeN},{mapDimN}]; (* map weight matrix *) :[font = input; preserveAspect; startGroup] i=0; iterN = 200; (* total number of iterations *) interN = 10; (* interval between displaying graph *) mapRate = .5; (* learning rate for map layer *) undercut = .01; (* amount below activation to set threshold *) thresholdV = Table [0,{inNodeN}]; (* threshold vaues for each unit *) (* threshold function *) thresh[aV_,tV_]:= MapThread[If[#1 >= #2, 1, 0]&, {aV,tV}]; inpoint = Table[0,{inDimN}]; (* initial input point for graphics*) indexV = Table[n,{n, inPatN}]; (* a table of indexes into the input matrix keeps track of the input order for graphing*) windexV = Table[n,{n, inPatN}]; (* windex keeps track of the winning node associated with each input for graphing *) winner = 1; :[font = message; inactive; preserveAspect] General::spell1: Possible spelling error: new symbol name "interN" is similar to existing symbol "iterN". :[font = message; inactive; preserveAspect; endGroup; endGroup] General::spell1: Possible spelling error: new symbol name "indexV" is similar to existing symbol "windexV". :[font = subsubsection; inactive; preserveAspect] Main Loop :[font = input; preserveAspect; startGroup; animationSpeed = 14; infiniteLoop; loopDistance = 21] Do [ showGraphics[]; Do [ (* If all inputs are used, reshuffle index and reuse. [ensures that all inputs presented once per block] *) indexV = If[ Mod [i,inPatN]==0, Sort [indexV, (rnd>0)&], indexV]; (* Select input point by incrementing through shuffled indexV. *) index = Mod [i,inPatN] +1; inpoint = inputM[[ indexV[[index ]] ]]; (* Calculate Input Activation *) inputActV = Map[activate[inpoint,#]&, inputWtsM]; (* Determine winner *) freeV = thresh[ inputActV, thresholdV]; winner = Flatten[Position[freeV*inputActV, Max[freeV*inputActV]]] [[1]]; inputActV [[winner]] = 1; (* Update input weights and threshold *) inputWtsM [[winner]] = inpoint; thresholdV[[winner]] = 1-undercut; (* Record index of winner associated with this input [for display] *) windexV [[ indexV[[index]] ]] = winner; (* Set map target to winner *) mapTarget = mapWtsM[[winner]]; (* Compute distances according to activations*) inputDistV = 1 - inputActV; mapDistV = 1 - Map[activate[mapTarget,#]&,mapWtsM]; (* Adjust map weights *) mapWtsM = mapWtsM + mapRate * (inputDistV - mapDistV) * 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0.80335 lineto 0.65126 0.73818 lineto 0.70794 0.61861 lineto 0.674 0.48217 lineto 0.56331 0.37453 lineto 0.43362 0.34894 lineto 0.3283 0.40411 lineto 0.27592 0.51152 lineto 0.29033 0.62958 lineto 0.35914 0.71518 lineto 0.45743 0.74841 lineto 0.54837 0.72806 lineto 0.59842 0.67278 lineto stroke grestore MathSubEnd grestore % End of sub-graphic grestore % End of Graphics MathPictureEnd :[font = subsection; inactive; preserveAspect; startGroup] References :[font = text; inactive; preserveAspect; endGroup; endGroup] Baird, J. C. & Noma, E. (1978). Fundamentals of scaling and psychophysics. New York: Wiley. Davidson, M. L. (1983). Multidimensional Scaling. New York: Wiley. Kohonen, T. (1982). Self-organized formation of topologically correct feature maps. Biological Cybernetics. 43: 59-69. ^*)