(************** Content-type: application/mathematica ************** Mathematica-Compatible Notebook This notebook can be used with any Mathematica-compatible application, such as Mathematica, MathReader or Publicon. The data for the notebook starts with the line containing stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). NOTE: If you modify the data for this notebook not in a Mathematica- compatible application, you must delete the line below containing the word CacheID, otherwise Mathematica-compatible applications may try to use invalid cache data. For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. *******************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 498940, 27153]*) (*NotebookOutlinePosition[ 499827, 27185]*) (* CellTagsIndexPosition[ 499783, 27181]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{Cell["\<\ from Exploring Artificial Neural Networks with Mathematica\[Trademark] \[Copyright] Jacques J. Vidal, 1991\ \>", "Subsubsection", ImageRegion->{{0, 1}, {0, 1}}], Cell[CellGroupData[{Cell["Continuous Units ", "Subsection", ImageRegion->{{0, 1}, {0, 1}}], Cell[CellGroupData[{Cell["Informal Problem Statement:", "Subsubsection", ImageRegion->{{0, 1}, {0, 1}}], Cell[TextData[{ StyleBox["The simplest formal \"neuron\" models are elemental processing \ units \nwith ", FontFamily->"Times"], StyleBox["n", FontFamily->"Times", FontSlant->"Italic"], StyleBox[" inputs and one single output. To generate this output, the \ unit first calculates a ", FontFamily->"Times"], StyleBox["weighted sum", FontFamily->"Times", FontSlant->"Italic"], StyleBox[" of the n inputs ", FontFamily->"Times"], StyleBox["{x1,x2...xn}", FontFamily->"Times", FontWeight->"Bold"], StyleBox[", a scalar function of the input vector. 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", FontFamily->"Times"] }], "Text", ImageRegion->{{0, 1}, {0, 1}}], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica MathPictureStart % Start of picture % Scaling calculations 0 1 0 1 [ [ 0.000000 0.000000 0 0 ] [ 1.000000 0.613821 0 0 ] ] MathScale % Start of Graphics 0 setgray 0 setlinewidth gsave 0.000000 0.000000 translate 1.000000 0.613821 scale 1 string 246 151 1 [246 0 0 151 0 0] { currentfile 1 index readhexstring pop } false 3 colorimage FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFC0 FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF8007 FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFC0 FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFC0FC0 FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF 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weight ", FontFamily->"Times"], StyleBox["w0", FontFamily->"Times", FontWeight->"Bold"], StyleBox[". \n ", FontFamily->"Times"], StyleBox["w0", FontFamily->"Times", FontWeight->"Bold"], StyleBox[" becomes a ", FontFamily->"Times"], StyleBox["threshold", FontFamily->"Times", FontSlant->"Italic"], StyleBox[" value when ", FontFamily->"Times"], StyleBox["discrete units", FontFamily->"Times", FontSlant->"Italic"], StyleBox[" with step functions are used. This extra input is useful when \ learning algorithms are considered. It allows the bias or threshold to be \ learned like any other weight. \n \nWe will start our exploration of \ Artificial Neural Networks with these continuous units.\nQ:", FontFamily->"Times"], StyleBox[" What is the utility of such a unit, by itself or in a network?", FontFamily->"Times", FontSlant->"Italic"], StyleBox["\nA: Each unit is an adjustable ", FontFamily->"Times"], StyleBox["function generator", FontFamily->"Times", FontSlant->"Italic"], StyleBox[", if we assume that the summing parameters, i.e., the weights \ can be varied. With the additional dummy input (-1), the value of the net \ input is the ", FontFamily->"Times"], StyleBox["Dot ", FontFamily->"Times", FontWeight->"Bold"], StyleBox["or", FontFamily->"Times"], StyleBox[" Inner", FontFamily->"Times", FontWeight->"Bold"], StyleBox[" product of the extended input vector with the weight vector. \ Both have the same dimension (n+1). If these vectors are normalized to unit \ length the net input equals to the cosine of the angle between the vectors, \ hence a measure of similarity between the input vector or \"pattern\" and the \ weight vector. With purely linear units, the processing of input data as \ well as the recipe for weight learning are analytically tractable. \ Considerable insight will be gained by examining the underlying theory as \ will be done in the next section.\nWhen a nonlinear transfer function is \ introduced, the output of each processing unit becomes constrained by the \ shape of this function. With a monotonous continuous function such as the \ previously mentioned sigmoid, it is easy to see how the unit becomes a ", FontFamily->"Times"], StyleBox["classifier", FontFamily->"Times", FontSlant->"Italic"], StyleBox[". This is clear when the transfer function is a thresholding \ step: the output can only assume one of two values (0,1) or (+1,-1) according \ to the sign of the net input. The dividing boundary is a hyperplane in the \ input space between the respective regions of positive and negative outputs . \ Hence the unit can classify data vectors (points) in the input space into \ two separate categories. The change of sign and the hyperplane boundary also \ holds with continuous sigmoid units, since sigmoid functions are linear \ around their threshold. In this case the value of the output is continuous \ through the boundary and approaches saturation at some distance from the \ hyperplane.\n\n", FontFamily->"Times"], StyleBox["Q: How does one find the weights that produce the right divider \ ?", FontFamily->"Times", FontSlant->"Italic"], StyleBox["\nA: This is the \"Learning\" problem. At this point we will be \ concerned with \"supervised learning\", i.e., learning from examples. The \ examples, are input vectors paired with a \"desired\" output. This is often \ called a \"training dataset\", or simply training set.\n\nWe will start with \ a purely linear transfer function to get a theoretical foothold on the \ problem. Then, if all the training data can be given in advance (batch), the \ problem has a theoretical solution from linear algebra. The ", FontFamily->"Times"], StyleBox["pseudo-inverse", FontFamily->"Times", FontSlant->"Italic"], StyleBox[" operator yields weights such that the input/output pairs \ produced by the unit match in the mean square sense, the data pairs in the \ training set.\nAn equivalent strategy is to follow the gradient over the \ error surface globally defined over the weight space by the training set. The \ surface is a paraboloid and gradient descent will therefore optimally lead to \ the (unique) minimum. Unlike the pseudo-inverse method, the gradient descent \ approach can be applied to nonlinear continuous units as well. The error \ surface however is specific to each specific training set and can only be \ correctly calculated if all the training pairs are used in calculating the \ error function.\n\nIf the training examples and weight corrections occur one \ at the time, we can still obtain an estimate of the control surface and use \ the gradient to guide us toward the minimum. This is sometine called the ", FontFamily->"Times"], StyleBox["stochastic approximation", FontFamily->"Times", FontSlant->"Italic"], StyleBox[" technique in system theory. The weights are learned by \ presenting the data repetitively, one at the time and typically several \ times over. This approach has robust properties, however convergence can no \ longer be guaranteed, a limitation that pervades all neural network methods.", FontFamily->"Times"] }], "Text", ImageRegion->{{0, 1}, {0, 1}}]}, Open]]}, Open]]}, Open]], Cell["Linear Units. The analytical approach:", "Subsubsection", ImageRegion->{{0, 1}, {0, 1}}], Cell[TextData[{ StyleBox["We first observe that if we were given three arbitrary \ input-output pairs produced by a linear unit, then these three points on the \ planar output suface uniquely define a plane. Since the parameters in the \ plane equation are the three weights, we should be able to recover their \ values\n by the ", FontFamily->"Times"], StyleBox["Inverse", FontFamily->"Times", FontWeight->"Bold"], StyleBox[" operator. Actually, the equation is invariant if these values \ are multiplied by a positive constant, but normalization will render them \ unique. ", FontFamily->"Times"] }], "Text", ImageRegion->{{0, 1}, {0, 1}}], Cell["Let us choose some arbitrary weights:", "Text", ImageRegion->{{0, 1}, {0, 1}}], Cell["weightVector = {0.5,0.7,-0.2};", "Input", ImageRegion->{{0, 1}, {0, 1}}], Cell["\<\ We will define a function that normalizes weight vectors to unit length. \ Normalization will facilitate later comparisons:\ \>", "Text", ImageRegion->{{0, 1}, {0, 1}}], Cell["normalize[p_List] := p/(Apply[Plus,p^2]^0.5);", "Input", InitializationCell->True, ImageRegion->{{0, 1}, {0, 1}}], Cell[CellGroupData[{Cell["{w0,w1,w2} = normalize[weightVector];", "Input", ImageRegion->{{0, 1}, {0, 1}}], Cell[CellGroupData[{Cell["weightVector =N[{w0,w1,w2},2] ", "Input", ImageRegion->{{0, 1}, {0, 1}}], Cell[OutputFormData["\<\ {0.5661385170722978753, 0.7925939239012170254, -0.2264554068289191501}\ \>", "\<\ {0.57, 0.79, -0.23}\ \>"], "Output", AspectRatioFixed->False, ImageRegion->{{0, 1}, {0, 1}}]}, Open]]}, Open]], Cell["\<\ To illustrate the process, let us generate the outputs generated with three \ arbitrary input pairs, (each augmented by the dummy input (-1) that allows \ for bias:\ \>", "Text", ImageRegion->{{0, 1}, {0, 1}}], Cell["\<\ inputVectors = {{-1.,-0.4,0.9},{-1,-0.7,-0.1},{-1,0.6,-0.2}};\t\t\ \>", "Input", ImageRegion->{{0, 1}, {0, 1}}], Cell["The outputs are simply the dot product with the weights :", "Text", ImageRegion->{{0, 1}, {0, 1}}], Cell[CellGroupData[{Cell["outputs = weightVector.Transpose[inputVectors]", "Input", ImageRegion->{{0, 1}, {0, 1}}], Cell[OutputFormData["\<\ {-1.086985952778811921, -1.098308723120257878, -0.04529108136578383008}\ \>", "\<\ {-1.08699, -1.09831, -0.0452911}\ \>"], "Output", ImageRegion->{{0, 1}, {0, 1}}]}, Open]], Cell[TextData[{ StyleBox[" Since we have three vector equations of the form ", FontFamily->"Times"], StyleBox["y=w.x", FontFamily->"Times", FontWeight->"Bold"], StyleBox[", and if we have the input vectors x properly arranged in a 3x3 \ matrix ", FontFamily->"Times"], StyleBox["X", FontFamily->"Times", FontWeight->"Bold"], StyleBox[" (this will require transposing ", FontFamily->"Times"], StyleBox["inputVectors", FontFamily->"Times", FontWeight->"Bold"], StyleBox["), we can write ", FontFamily->"Times"], StyleBox["w = y.X^(-1)", FontFamily->"Times", FontWeight->"Bold"], StyleBox[". Hence, applying the ", FontFamily->"Times"], StyleBox["Inverse", FontFamily->"Times", FontWeight->"Bold"], StyleBox[" operator, we should recover the weightVector:", FontFamily->"Times"] }], "Text", ImageRegion->{{0, 1}, {0, 1}}], Cell[CellGroupData[{Cell["\<\ computedWeightVector=N[outputs.Inverse[Transpose[inputVectors]],2]\ \>", "Input", ImageRegion->{{0, 1}, {0, 1}}], Cell[OutputFormData["\<\ {0.5661385170722978754, 0.7925939239012170254, -0.22645540682891915}\ \>", "\<\ {0.57, 0.79, -0.23}\ \>"], "Output", ImageRegion->{{0, 1}, {0, 1}}]}, Open]], Cell[TextData[{ StyleBox["The result should match the previously chosen ", FontFamily->"Times"], StyleBox["weightVector", FontFamily->"Times", FontWeight->"Bold"], StyleBox["; this can be checked by showing that the difference between the \ two vectors is equal to or is very close to zero.", FontFamily->"Times"] }], "Text", ImageRegion->{{0, 1}, {0, 1}}], Cell[CellGroupData[{Cell["Chop[computedWeightVector-weightVector]", "Input", ImageRegion->{{0, 1}, {0, 1}}], Cell[OutputFormData["\<\ {0, 0, 0}\ \>", "\<\ {0, 0, 0}\ \>"], "Output", ImageRegion->{{0, 1}, {0, 1}}]}, Open]], Cell["\<\ Linear Regression: Calculating Best Weights over a set of sample points\ \>", "Subsubsection", ImageRegion->{{0, 1}, {0, 1}}], Cell[TextData[{ StyleBox["With three input vectors, the weight identification problem \ solves exactly. It is neither overconstrained or underconstrained since there \ is one and only one plane passing throught three points, unless the points \ are colinear.\nWe now will generalize this analytical solution to the \ overconstrained case, i.e., when there is a set of input vectors and \ \"desired outputs\" and the best approximating plane is sought, in the space \ defined by the input-output pairs. In other words, given a set of \ input-output pairs, we will seek to calculate the weights that generate an \ \"optimal\" plane through the points defined by the input vectors and \ corresponding outputs. \nThis is also called \"linear regression\": finding \ the best plane that fits data vectors when there are more than (n+1) points \ where n is the dimensionality of the input vectors. The data points need \ not, and generally will not fall on the resulting plane and may not even form \ an approximate planar arrangement.\nA theoretical solution will be obtained \ if we interpret \"optimal\" as meaning that the plane defined by the optimal \ weights found is such that the mean square distance from all data points to \ the plane is minimized (Least Mean Squared or LMS).\n\n Let us first pick up \ ", FontFamily->"Times"], StyleBox["numPoints", FontFamily->"Times", FontWeight->"Bold"], StyleBox[" data points at random and plot the result:", FontFamily->"Times"] }], "Text", ImageRegion->{{0, 1}, {0, 1}}], Cell[CellGroupData[{Cell["\<\ numPoints = 20; u := 2 Random[]-1; inputVectors = Table [ {-1, u,u},{numPoints}];\t\t\t\t\t\t\ \>", "Input", ImageRegion->{{0, 1}, {0, 1}}], Cell["inputs=Map[Drop[#,1]&,inputVectors];", "Input", ImageRegion->{{0, 1}, {0, 1}}]}, Open]], Cell["\<\ We will now create output data. First we will generate the values that fall \ exactly on the planar surface defined by the previously chosen set of \ weights:\ \>", "Text", ImageRegion->{{0, 1}, {0, 1}}], Cell[CellGroupData[{Cell["Short[outputs = weightVector.Transpose[inputVectors]]", "Input", ImageRegion->{{0, 1}, {0, 1}}], Cell[OutputFormData["\<\ Short[{-0.226125087087284076, -0.1497662497619224047, -0.0003532536433389931832, -0.1873679774112510604, -0.2187208999304635048, -0.7323607467690519989, 0.005579359343085727948, 0.1362814084115510845, 0.1526275177859386972, 0.2557521661374974597, -0.8526148331995216009, 0.05498805720449910002, -0.2418041085541893194, 0.01976431399746991313, 0.09625076264945453593, -0.3850584563662493051, -0.7387233029680591908, -0.9901809846066393088, 0.1717843130737924486, -1.097029989581674474, -0.08409762339873233993, -1.060813075013355948, -0.4434134582257955568, -0.9299886452120584316, -0.8998942132066699545, -0.144123498444875694, -0.001690168079895732811, -0.7907871638954453953, -0.6226683990395172917, -0.07829114813580408278, -0.547085167797284344, -0.5900470127850962669, -0.5812218494633516707, -0.5282704960822314508, -1.399458953726297967, -1.309270003131075202, -0.2108885606256534908, -0.9276572852327779709, -0.486854249652366237, -0.2188649745742174237, -0.2722934724578510538, 0.2502688123082346603, -1.273680020205772924, -1.064554373559454369, -1.137850741727547351, -0.6599340288550498878, -0.8039469967636623619, -1.092925335681282113, -0.9062871242124176541, -0.2702960229072524818}]\ \>", "\<\ {-0.226125, -0.149766, -0.000353254, <<45>>, -0.906287, -0.270296}\ \>"], "Output", AspectRatioFixed->False, ImageRegion->{{0, 1}, {0, 1}}]}, Open]], Cell[TextData[{ StyleBox["Linear Algebra tells us that the best linear match to these can \ be calculated as previously by using the ", FontFamily->"Times"], StyleBox["PseudoInverse", FontFamily->"Times", FontWeight->"Bold"], StyleBox[" operator. At this time, since we have outputs that fall exactly \ on the theoretical plane, we should exactly or almost exactly recover the \ weightVector. The Pseudo-Inverse works in this case like the Inverse did \ earlier:", FontFamily->"Times"] }], "Text", ImageRegion->{{0, 1}, {0, 1}}], Cell[CellGroupData[{Cell["\<\ computedWeightVector = outputs.PseudoInverse[Transpose[inputVectors]]\t\t\ \>", "Input", ImageRegion->{{0, 1}, {0, 1}}], Cell[OutputFormData["\<\ {0.5661385170722978752, 0.7925939239012170255, -0.2264554068289191502}\ \>", "\<\ {0.566139, 0.792594, -0.226455}\ \>"], "Output", AspectRatioFixed->False, ImageRegion->{{0, 1}, {0, 1}}]}, Open]], Cell["and indeed", "Text", AspectRatioFixed->False, ImageRegion->{{0, 1}, {0, 1}}], Cell[CellGroupData[{Cell["Chop[computedWeightVector-weightVector]\t", "Input", ImageRegion->{{0, 1}, {0, 1}}], Cell[OutputFormData["\<\ {0, 0, 0}\ \>", "\<\ {0, 0, 0}\ \>"], "Output", AspectRatioFixed->False, ImageRegion->{{0, 1}, {0, 1}}]}, Open]], Cell["\<\ To model a more realistic situation, we now add some noise to scatter the \ desired outputs somewhat, although remaining in the vicinity of the plane:\ \>", "Text", ImageRegion->{{0, 1}, {0, 1}}], Cell["noiseList = Table[ Random[]/10,{numPoints} ];", "Input", ImageRegion->{{0, 1}, {0, 1}}], Cell[CellGroupData[{Cell["Short[outputs = outputs + noiseList]", "Input", ImageRegion->{{0, 1}, {0, 1}}], Cell[OutputFormData["\<\ Short[{-0.1497395645256425686, -0.121862846015093228, 0.08286993767027263928, -0.1306451273808907696, -0.1211766709142700052, -0.6559683968641376498, 0.02024515076362491881, 0.2091661674876637174, 0.2506790404363776204, 0.3233539183254867689, -0.7833939396453077754, 0.06873983110370008532, -0.1667618736686551422, 0.07761871234580448159, 0.1921127182551973845, -0.2986694795492198495, -0.7044721044218063394, -0.894163563552532738, 0.2169892588670494127, -1.034795192458117346, -0.08326831900913391295, -0.9787178443732458119, -0.3922974842567161344, -0.9055254280730379579, -0.8213294533376067217, -0.1422314018205810837, 0.01480425568939739079, -0.7906776195362259389, -0.5744819770298798161, -0.01464107514821085374, -0.4994504600568307309, -0.5223120361222620506, -0.5361069572440034376, -0.4741085655003003189, -1.35808825882952426, -1.300184726613752528, -0.1312972622640237064, -0.8895446929145025099, -0.4058399727270332001, -0.190323047228864735, -0.2031998441888205225, 0.3114558624891873798, -1.21462833684267257, -1.000664280543736749, -1.068620066399827948, -0.5944687608175299559, -0.7690101833694486123, -1.009758783116716219, -0.8129529099275211583, -0.1779315449550150167}]\ \>", "\<\ {-0.14974, -0.121863, 0.0828699, <<8>>5, <<44>>, -0.812953, -0.177932}\ \>"], "Output", AspectRatioFixed->False, ImageRegion->{{0, 1}, {0, 1}}]}, Open]], Cell["\<\ Aeta, applying the pseudo-inverse operator, we should get the LMS weight \ vector, which should not be very far from the original since the points still \ lie near the original plane.\ \>", "Text", ImageRegion->{{0, 1}, {0, 1}}], Cell[CellGroupData[{Cell["\<\ computedWeightVector=outputs.PseudoInverse[Transpose[inputVectors]]\t\t\ \>", "Input", ImageRegion->{{0, 1}, {0, 1}}], Cell[OutputFormData["\<\ {0.5096955897765560998, 0.7916894180759024584, -0.2389143767084971252}\ \>", "\<\ {0.509696, 0.791689, -0.238914}\ \>"], "Output", AspectRatioFixed->False, ImageRegion->{{0, 1}, {0, 1}}]}, Open]], Cell["\<\ We can now check the effect of the added noise by comparing the recovered \ weight vector with the original:\ \>", "Text", ImageRegion->{{0, 1}, {0, 1}}], Cell[CellGroupData[{Cell["computedWeightVector-weightVector", "Input", ImageRegion->{{0, 1}, {0, 1}}], Cell[OutputFormData["\<\ {-0.05644292729574177552, -0.0009045058253145670685, -0.01245896987957797501}\ \>", "\<\ {-0.0564429, -0.000904506, -0.012459}\ \>"], "Output", AspectRatioFixed->False, ImageRegion->{{0, 1}, {0, 1}}]}, Open]], Cell[CellGroupData[{Cell["\<\ Lyapunov Functions, Error Gradients and Steepest Descent: Iterative \ Regression.\ \>", "Subsubsection", ImageRegion->{{0, 1}, {0, 1}}], Cell[TextData[{ StyleBox["Q: Since weights can be calculated, why bother with learning?", FontFamily->"Times", FontSlant->"Italic"], StyleBox["\nA: The previous method is usually not applicable. \nThe first \ reason is that the pseudo-inverse approach is a batch method, i.e., you must \ have all the data at once. In a typical learning situation, data comes one \ point at a time, and one will want to get successive approximations of the \ optimal weights. The second reason is of course that this batch method works \ only in the linear case. It will not accommodate units that have a nonlinear \ transfer function such as the sigmoid.\n\nLearning can be stated as an \ optimization problem. If we can find a global cost function of the weights \ whose minimization leads to the best set of weights (a Lyapunov function), \ we have a basis for successive approximation (the steepest descent). A \ natural candidate is quadratic function, the mean square of the difference \ between ", FontFamily->"Times"], StyleBox["desired", FontFamily->"Times", FontSlant->"Italic"], StyleBox[" (given in the training set) and ", FontFamily->"Times"], StyleBox["observed", FontFamily->"Times", FontSlant->"Italic"], StyleBox[" outputs, summed over the whole data set. To calculate this \ function aeta requires that all the training data be available at the onset. \ In the linear case, it can be shown that the pseudo-inverse operator finds \ weights that minimize this particular function. ", FontFamily->"Times"] }], "Text", ImageRegion->{{0, 1}, {0, 1}}], Cell[TextData[{ StyleBox["The shape of this error function is that a hyperparaboloid with a \ unique minimum. Its ellipticity is affected by the amount of covariance \ between input pairs. We can check the statistics, represented by the \ covariance matrix, of the input data above. First we observe that, for a \ uniform distribution over the {0,1} range, the Expected value of the variance \ (diagonal) terms is 1/3 while the covariance terms should be zero. \nOf \ course we should expect to be somewhat off from these theoretical values \ because 1) we have only a few points and 2) the pseudo-random generator is \ bound to generate unwanted correlations. \nMultiplying the ( ", FontFamily->"Times"], StyleBox["numPoints", FontFamily->"Times", FontWeight->"Bold"], StyleBox[" by 2) matrix formed from ", FontFamily->"Times"], StyleBox["inputVectors", FontFamily->"Times", FontWeight->"Bold"], StyleBox[" by its transpose and normalizing will yield the data correlation \ matrix. ", FontFamily->"Times"] }], "Text", ImageRegion->{{0, 1}, {0, 1}}], Cell[CellGroupData[{Cell["\<\ TableForm [(Transpose[inputs].inputs)/numPoints] \ \>", "Input", ImageRegion->{{0, 1}, {0, 1}}], Cell[OutputFormData["\<\ TableForm[{{0.3575869461283258068, 0.07380158886393323868}, {0.07380158886393323868, 0.3751886828895517923}}]\ \>", "\<\ 0.357587 0.0738016 0.0738016 0.375189\ \>"], "Output", AspectRatioFixed->False, ImageRegion->{{0, 1}, {0, 1}}]}, Open]], Cell[TextData[{ StyleBox["The error function is a scalar function of the weight vectors . \ We will make it visualizable by looking only at the dependency on ", FontFamily->"Times"], StyleBox["w1", FontFamily->"Times", FontWeight->"Bold"], StyleBox[" and", FontFamily->"Times"], StyleBox[" w2", FontFamily->"Times", FontWeight->"Bold"], StyleBox[". We will leave the bias ", FontFamily->"Times"], StyleBox["w0", FontFamily->"Times", FontWeight->"Bold"], StyleBox[" fixed to the value just found. (It should be close to zero \ anyway since the data generator is symmetrical in the +1,-1 range.)", FontFamily->"Times"] }], "Text", ImageRegion->{{0, 1}, {0, 1}}], Cell["Clear[costFunction1,w1,w2,k]", "Input", ImageRegion->{{0, 1}, {0, 1}}], Cell[TextData[{ StyleBox["The cost is the sum of the squared differences between the given \ (desired) outputs and the computed output, expressed as a function of the \ current values of the weights ", FontFamily->"Times"], StyleBox["w1", FontFamily->"Times", FontWeight->"Bold"], StyleBox[" and ", FontFamily->"Times"], StyleBox["w2", FontFamily->"Times", FontWeight->"Bold"], StyleBox[". Here is an expression for it, using the ", FontFamily->"Times"], StyleBox["Sum", FontFamily->"Courier", FontSize->14, FontWeight->"Bold"], StyleBox[" primitive. ", FontFamily->"Times"] }], "Text", ImageRegion->{{0, 1}, {0, 1}}], Cell[CellGroupData[{Cell["\<\ costFunction1 [w1_,w2_] = Simplify[ (0.5/ numPoints)* Sum[((outputs[[k]] - ({w0,w1,w2} .inputVectors[[k]])))^2, {k,numPoints}]] \ \>", "Input", ImageRegion->{{0, 1}, {0, 1}}], Cell["\<\ Here is another form for the same expression. It uses the parallel \ primitives Apply and Map and is more elegant: \ \>", "Text", ImageRegion->{{0, 1}, {0, 1}}], Cell["Clear[costFunction2,w1,w2]", "Input", ImageRegion->{{0, 1}, {0, 1}}], Cell[CellGroupData[{Cell["\<\ costFunction2 [w1_,w2_] = Simplify[ (0.5/ numPoints)* Apply[Plus,(outputs - Map[({w0,w1,w2}.#)&,inputVectors])^2]] \ \>", "Input", ImageRegion->{{0, 1}, {0, 1}}], Cell[OutputFormData["\<\ 0.1156358749024709763 - 0.2705288323279517282*w1 + 0.1787934730641629034*w1^2 + 0.03488852070614827756*w2 + 0.07380158886393323869*w1*w2 + 0.1875943414447758961*w2^2\ \>", "\<\ 2 0.115636 - 0.270529 w1 + 0.178793 w1 + 0.0348885 w2 + 0.0738016 w1 w2 + 2 0.187594 w2\ \>"], "Output", AspectRatioFixed->False, ImageRegion->{{0, 1}, {0, 1}}], Cell["We can verify that indeed we get the same formal expression", "Text", ImageRegion->{{0, 1}, {0, 1}}], Cell[CellGroupData[{Cell["SameQ[costFunction1[w1_,w2_],costFunction2[w1_,w2_]]", "Input", ImageRegion->{{0, 1}, {0, 1}}], Cell[OutputFormData["\<\ True\ \>", "\<\ True\ \>"], "Output", AspectRatioFixed->False, ImageRegion->{{0, 1}, {0, 1}}]}, Open]]}, Open]]}, Open]], Cell["\<\ w1Low=w2Low=-10; w1Up=w2Up=10;\ \>", "Input", AspectRatioFixed->False, ImageRegion->{{0, 1}, {0, 1}}], Cell["Clear [w1,w2];", "Input", AspectRatioFixed->False, ImageRegion->{{0, 1}, {0, 1}}], Cell[CellGroupData[{Cell["\<\ Plot3D [ costFunction2[w1,w2], \t{w1,w1Low, w1Up }, \t{w2,w2Low, w2Up },AxesLabel->{\"w1\",\"w2\",\"cost \"} ] \ \>", "Input", Evaluatable->False, ImageRegion->{{0, 1}, {0, 1}}], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: 0.82055 MathPictureStart /Courier findfont 10 scalefont setfont % Scaling calculations 0.0249355 0.99742 -0.0396341 0.99742 [ [(-10)] 0.05113 0.25884 1 0.933946 Msboxa [(-5)] 0.1926 0.20316 0.96648 1 Msboxa [(0)] 0.34275 0.1441 0.862234 1 Msboxa [(5)] 0.50241 0.08133 0.757988 1 Msboxa [(10)] 0.6725 0.0145 0.653742 1 Msboxa [(w1)] 0.30204 0.09689 0.862234 1 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If the ", FontFamily->"Times"], StyleBox["inputVectors", FontFamily->"Times", FontWeight->"Bold", FontSlant->"Italic"], StyleBox[" are also binary a threshold unit implements a Boolean function.\n\ ", FontFamily->"Times"], StyleBox["Q: What is a mathematical expression for the Sigmoid?", FontFamily->"Times", FontSlant->"Italic"], StyleBox["\nA: Different forms are used. A good one is the hyperbolic \ tangent which is bounded by -1 and +1. 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} ] *) (******************************************************************* End of Mathematica Notebook file. *******************************************************************)