# Example 6.4: Regressor selection problem

Jump to: Source code Text output Plots Library index

    % Section 6.3.1, Figure 6.7
    % Original by Lieven Vandenberghe
    % Adapted for CVX Argyris Zymnis - 10/2005
    %
    % Solves
    %        minimize   ||A*x-b||_2
    %        subject to card(x) <= k
    %
    % where card(x) denotes the number of nonzero elements in x,
    % by first solving (for some value of alpha close to ||x_ln||_1)
    %        minimize   ||A*x-b||_2
    %        subject to ||x||_1 <= alpha
    %
    % and iteratively decreasing alpha so as to get card(x) = k
    % The sparsity pattern is then fixed in A and b and
    %        minimize   ||A*x-b||_2
    %
    % is solved

    rand('state',0);

    m = 10;
    n = 20;

    A = randn(m,n);
    b = A*[randn(round(m/2),1); zeros(n-round(m/2),1)];
    b = b + 0.1*norm(b)*randn(m,1);

    if (1) %%%%%%%%%%%%

    % tradeoff curve for heuristic
    %
    % min.  ||Ax-b||_2
    % s.t.  ||x||_1 <= alpha

    residuals_heur = [norm(b)];
    xln = A'*((A*A')\b);
    lnorm = norm(xln,1);
    nopts = 100;
    alphas = linspace(0,lnorm,nopts);
    residuals_heur = [norm(b)];
    card_heur = [0];

    for k=2:(nopts-1)
      alpha = alphas(k);

      cvx_begin quiet
        variable x(n)
        minimize(norm(A*x-b))
        subject to
            norm(x,1) <= alpha;
      cvx_end

      x(find(abs(x) < 1e-3*max(abs(x)))) = 0;
      ind = find(abs(x));
      sparsity = length(ind);
      fprintf(1,'Current sparsity pattern k = %d \n',sparsity);
      x = zeros(n,1);  x(ind) = A(:,ind)\b;
      card_heur = [card_heur, sparsity];
      residuals_heur = [residuals_heur, norm(A*x-b)];
    end;

    obj1 = norm(b)
    obj2 = [0];

    i=1;
    for k=1:m-1
      if ~isempty(find(card_heur == k))
         obj2(i+1) = k;
         obj1(i+1) = min(residuals_heur(find(card_heur ==k)));
         i=i+1;
      end;
    end;
    obj2(i) = m;  obj1(i) = 0;

    end; %%%%%%%%%%%%%%%%%%%

    % globally optimal tradeoff

    if (1) %%%%%%%%%%%%%

    bestx = zeros(n,m);
    bestres = zeros(1,m);

    for k=1:m-1
      k
      % enumerate sparsity patterns with exactly k nonzeros
      bestres(k) = Inf;
      ind = 1:k
      nocases = 1;
      done = 0;
      while ~done
         done = 1;
         for i=0:k-1
           if (ind(k-i) < n-i),
              ind(k-i:k) = ind(k-i)+[1:i+1];
              done = 0;
              break;
           end;
         end;
         if done, break; end;
         x = zeros(n,1);
         x(ind) = A(:,ind)\b;
         if (norm(A*x-b) < bestres(k)),
            bestres(k) = norm(A*x-b);
            bestx(:,k) = x;
         end;
         nocases = nocases + 1;
      end;
      nocases
      factorial(n)/(factorial(n-k)*factorial(k))
    end;

    x = A\b;
    bestres(m) = norm(A*x-b);
    bestres = [norm(b) bestres];

    end; %%%%%%%%%

    figure
    hold off
    obj1dbl =[];
    obj2dbl =[];
    for i=1:length(obj1)-1
      obj1dbl = [obj1dbl, obj1(i), obj1(i)];
      obj2dbl = [obj2dbl, obj2(i), obj2(i+1)];
    end;
    obj1dbl = [obj1dbl, obj1(length(obj1))];
    obj2dbl = [obj2dbl, obj2(length(obj1))];

    bestobj1 = bestres;
    bestobj2 = [0:1:m];
    bestobj1dbl =[];
    bestobj2dbl =[];
    for i=1:length(bestobj1)-1
      bestobj1dbl = [bestobj1dbl, bestobj1(i), bestobj1(i)];
      bestobj2dbl = [bestobj2dbl, bestobj2(i), bestobj2(i+1)];
    end;
    bestobj1dbl = [bestobj1dbl, bestobj1(length(bestobj1))];
    bestobj2dbl = [bestobj2dbl, bestobj2(length(bestobj1))];

    plot(obj1dbl,obj2dbl,'-', bestobj1dbl, bestobj2dbl,'--');
    hold on
    plot(obj1,obj2,'o', bestobj1, bestobj2,'o');
    axis([0 ceil(2*norm(b))/2 0 m+1])
    xlabel('x');
    ylabel('y');
    hold off

    %print -deps sparse_regressor_global_helv.eps
    %save regressor_results

    Current sparsity pattern k = 1
    Current sparsity pattern k = 1
    Current sparsity pattern k = 1
    Current sparsity pattern k = 1
    Current sparsity pattern k = 1
    Current sparsity pattern k = 1
    Current sparsity pattern k = 1
    Current sparsity pattern k = 1
    Current sparsity pattern k = 1
    Current sparsity pattern k = 1
    Current sparsity pattern k = 1
    Current sparsity pattern k = 1
    Current sparsity pattern k = 1
    Current sparsity pattern k = 1
    Current sparsity pattern k = 1
    Current sparsity pattern k = 1
    Current sparsity pattern k = 1
    Current sparsity pattern k = 1
    Current sparsity pattern k = 2
    Current sparsity pattern k = 2
    Current sparsity pattern k = 2
    Current sparsity pattern k = 2
    Current sparsity pattern k = 2
    Current sparsity pattern k = 2
    Current sparsity pattern k = 2
    Current sparsity pattern k = 2
    Current sparsity pattern k = 2
    Current sparsity pattern k = 2
    Current sparsity pattern k = 2
    Current sparsity pattern k = 3
    Current sparsity pattern k = 3
    Current sparsity pattern k = 4
    Current sparsity pattern k = 4
    Current sparsity pattern k = 4
    Current sparsity pattern k = 4
    Current sparsity pattern k = 4
    Current sparsity pattern k = 4
    Current sparsity pattern k = 4
    Current sparsity pattern k = 4
    Current sparsity pattern k = 5
    Current sparsity pattern k = 5
    Current sparsity pattern k = 5
    Current sparsity pattern k = 5
    Current sparsity pattern k = 5
    Current sparsity pattern k = 5
    Current sparsity pattern k = 5
    Current sparsity pattern k = 6
    Current sparsity pattern k = 7
    Current sparsity pattern k = 8
    Current sparsity pattern k = 7
    Current sparsity pattern k = 7
    Current sparsity pattern k = 7
    Current sparsity pattern k = 9
    Current sparsity pattern k = 10
    Current sparsity pattern k = 10
    Current sparsity pattern k = 10
    Current sparsity pattern k = 10
    Current sparsity pattern k = 10
    Current sparsity pattern k = 10
    Current sparsity pattern k = 10
    Current sparsity pattern k = 9
    Current sparsity pattern k = 9
    Current sparsity pattern k = 8
    Current sparsity pattern k = 8
    Current sparsity pattern k = 8
    Current sparsity pattern k = 8
    Current sparsity pattern k = 8
    Current sparsity pattern k = 9
    Current sparsity pattern k = 10
    Current sparsity pattern k = 12
    Current sparsity pattern k = 14
    Current sparsity pattern k = 15
    Current sparsity pattern k = 16
    Current sparsity pattern k = 19
    Current sparsity pattern k = 18
    Current sparsity pattern k = 18
    Current sparsity pattern k = 18
    Current sparsity pattern k = 19
    Current sparsity pattern k = 19
    Current sparsity pattern k = 19
    Current sparsity pattern k = 19
    Current sparsity pattern k = 18
    Current sparsity pattern k = 18
    Current sparsity pattern k = 19
    Current sparsity pattern k = 19
    Current sparsity pattern k = 20
    Current sparsity pattern k = 19
    Current sparsity pattern k = 20
    Current sparsity pattern k = 20
    Current sparsity pattern k = 20
    Current sparsity pattern k = 19
    Current sparsity pattern k = 20
    Current sparsity pattern k = 19
    Current sparsity pattern k = 19
    Current sparsity pattern k = 20
    Current sparsity pattern k = 20
    Current sparsity pattern k = 20
    Current sparsity pattern k = 20

    obj1 =

        2.8954

    k =

         1

    ind =

         1

    nocases =

        20

    ans =

        20

    k =

         2

    ind =

         1     2

    nocases =

       190

    ans =

       190

    k =

         3

    ind =

         1     2     3

    nocases =

            1140

    ans =

            1140

    k =

         4

    ind =

         1     2     3     4

    nocases =

            4845

    ans =

            4845

    k =

         5

    ind =

         1     2     3     4     5

    nocases =

           15504

    ans =

           15504

    k =

         6

    ind =

         1     2     3     4     5     6

    nocases =

           38760

    ans =

           38760

    k =

         7

    ind =

         1     2     3     4     5     6     7

    nocases =

           77520

    ans =

           77520

    k =

         8

    ind =

         1     2     3     4     5     6     7     8

    nocases =

          125970

    ans =

          125970

    k =

         9

    ind =

         1     2     3     4     5     6     7     8     9

    nocases =

          167960

    ans =

          167960