1 function sheet04
   2 
   3 % get the data
   4 [X, Y] = twomoon(1000);
   5 
   6 % get a subset of the data
   7 figure(1)
   8 NS = 10;
   9 [XS, YS] = subset(NS, X, Y);
  10 plot2ddata(X, Y);
  11 hold on
  12 plot(XS(:, 1), XS(:, 2), 'kx', 'MarkerSize', 10, 'LineWidth', 3);
  13 hold off
  14 title('data and example points')
  15 
  16 % train KRR
  17 figure(2);
  18 K = rbfkern(1, XS, XS);
  19 alpha = inv(K + 1e-6*eye(NS))*YS;
  20 
  21 YH = rbfkern(1, X, XS) * alpha;
  22 
  23 plot2ddata(X, YH);
  24 hold on;
  25 plot(XS(:,1), XS(:, 2), 'kx', 'MarkerSize', 10, 'LineWidth', 3);
  26 hold off
  27 
  28 title('learned only using sample points')
  29 
  30 % train with KRR + Graph Laplacian regularization
  31 figure(3)
  32 alpha = trainGLKRR(1, 0.001, 0.1, 1, XS, YS, X);
  33 YH = rbfkern(1, X, X) * alpha;
  34 plot2ddata(X, YH);
  35 hold on;
  36 plot(XS(:,1), XS(:, 2), 'kx', 'MarkerSize', 10, 'LineWidth', 3);
  37 hold off
  38 
  39 title('learned with graph Laplacian regularizer')
  40 
  41 function K = rbfkern(w, X, Y)
  42 N = size(X, 1);
  43 M = size(Y, 1);
  44 XX = sum(X.*X, 2);
  45 YY = sum(Y.*Y, 2);
  46 D = repmat(XX, 1, M) + repmat(YY', N, 1) - 2 * X * Y';
  47 K = exp(-D/(2*w));
  48 
  49 function plot2ddata(X, Y)
  50 P = (sign(Y) == 1);
  51 N = (sign(Y) == -1);
  52 plot(X(P, 1), X(P, 2), 'r+', X(N, 1), X(N, 2), 'bo');
  53 
  54 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  55 %
  56 % your solution below
  57 %
  58 
  59 % 1. Generate a two-moon data set with N points and some noise. Make sure
  60 % that classes are not linearly seprable.
  61 function [X, Y] = twomoon(N)
  62 % ...
  63 
  64 % 2. Choose random subset of size N from a data set X, Y.
  65 function [X, Y] = subset(N, X, Y)
  66 % ...
  67 
  68 % 3. Compute the graph-Laplacian for Gaussian weights with width w
  69 function L = graphLaplacian(w, X)
  70 % ...
  71 
  72 % 4. Train KRR with Graph-Laplacian regularization on the data
  73 %
  74 % Parameters:
  75 %    w - width for the rbf kernel
  76 %    g - width for the graph-Laplacian
  77 %    lambda - regularization constant for general smoothness
  78 %    tau - regularization constant for GL-regularizer
  79 %    XS, YS - data set with labels
  80 %    X - unlabeled data
  81 function alpha = trainGLKRR(w, g, lambda, tau, XS, YS, X)
  82 % ...