Langrangian SVM的使用~·
转自：http://bbs.sciencenet.cn/forum.php?mod=viewthread&tid=523129
http://research.cs.wisc.edu/dmi/lsvm/
function [iter, optCond, time, w, gamma] = lsvm(A,D,nu,tol,maxIter,alpha,
...
perturb,normalize);
% LSVM Langrangian Support Vector Machine algorithm
% LSVM solves a support vector machine problem using an iterative
% algorithm inspired by an augmented Lagrangian formulation.
%
% iters = lsvm(A,D)
%
% where A is the data matrix, D is a diagonal matrix of 1s and -1s
% indicating which class the points are in, and 'iters' is the number
% of iterations the algorithm used.
%
% All the following additional arguments are optional:
%
% [iters, optCond, time, w, gamma] = ...
% lsvm(A,D,nu,tol,maxIter,alpha,perturb,normalize)
%
% optCond is the value of the optimality condition at termination.
% time is the amount of time the algorithm took to terminate.
% w is the vector of coefficients for the separating hyperplane.
% gamma is the threshold scalar for the separating hyperplane.
%
% On the right hand side, A and D are required. If the rest are not
% specified, the following defaults will be used:
% nu = 1/size(A,1), tol = 1e-5, maxIter = 100, alpha = 1.9/nu,
% perturb = 0, normalize = 0
%
% perturb indicates that all the data should be perturbed by a random
% amount between 0 and the value given. perturb is recommended only
% for highly degenerate cases such as the exclusive or.
%
% normalize should be set to 1 if the data should be normalized before
% training.
%
% The value -1 can be used for any of the entries (except A and D) to
% specify that default values should be used.
%
% Copyright (C) 2000 Olvi L. Mangasarian and David R. Musicant.
% Version 1.0 Beta 1
% This software is free for academic and research use only.
% For commercial use, contact [email protected].
% If D is a vector, convert it to a diagonal matrix.
[k,n] = size(D);
if k==1 | n==1
D=diag(D);
end;
% Check all components of D and verify that they are +-1
checkall = diag(D)==1 | diag(D)==-1;
if any(checkall==0)
error('Error in D: classes must be all 1 or -1.');
end;
m = size(A,1);
if ~exist('nu') | nu==-1
nu = 1/m;
end;
if ~exist('tol') | tol==-1
tol = 1e-5;
end;
if ~exist('maxIter') | maxIter==-1
maxIter = 100;
end;
if ~exist('alpha') | alpha==-1
alpha = 1.9/nu;
end;
if ~exist('normalize') | normalize==-1
normalize = 0;
end;
if ~exist('perturb') | perturb==-1
perturb = 0;
end;
% Do a sanity check on alpha
if alpha > 2/nu,
disp(sprintf('Alpha is larger than 2/nu. Algorithm may not converge.'));
end;
% Perturb if appropriate
rand('seed',22);
if perturb,
A = A + rand(size(A))*perturb;
end;
% Normalize if appropriate
if normalize,
avg = mean(A);
dev = std(A);
if (isempty(find(dev==0)))
A = (A - avg(ones(m,1), : ) )./dev(ones(m,1), : ) ;
else
warning('Could not normalize matrix: at least one column is constant.')
;
end;
end;
% Create matrix H
[m,n] = size(A);
e = ones(m,1);
H = D*[A -e];
iter = 0;
time = cputime;
% "K" is an intermediate matrix used often in SMW calclulations
K = H*inv((speye(n+1)/nu+H'*H));
% Choose initial value for x
x = nu*(1-K*(H'*e));
% y is the old value for x, used to check for termination
y = x + 1;
while iter < maxIter & norm(y-x)>tol
% Intermediate calculation which is used twice
z = (1+pl(((x/nu+H*(H'*x))-alpha*x)-1));
y = x;
% Calculate new value of x
x=nu*(z-K*(H'*z));
iter = iter + 1;
end;
% Determine outputs
time = cputime - time;
optCond = norm(x-y);
w = A'*D*x;
gamma = -e'*D*x;
disp(sprintf('Running time (CPU secs) = %g',time));
disp(sprintf('Number of iterations = %d',iter));
disp(sprintf('Training accuracy = %g',sum(D*(A*w-gamma)>0)/m));
return;
function pl = pl(x);
%PLUS function : max{x,0}
pl = (x+abs(x))/2;
return;
举例：A =
41 250
42 238
26 196
63 368
45 350
55 316
57 400
66 402
64 424
38 254
40 350
54 318
55 348
54 362
59 428
74 452
63 468
70 484
56 366
38 238
45 328
61 418
91 576
61 346
66 462
50 292
63 430
51 350
24 170
64 410
64 410
53 342
43 270
79 602
44 332
79 536
58 392
68 392
18 104
64 434
59 390
40 246
40 228
43 364
89 602
23 146
47 308
76 492
52 348
89 520
76 596
77 576
81 624
74 580
70 482
70 576
76 616
52 354
42 272
81 634
56 446
102 696
86 584
88 694
109 826
57 468
61 614
61 532
61 430
101 796
96 850
64 526
94 798
51 342
61 538
68 586
84 710
127 974
93 700
94 730
90 680
66 558
66 598
85 748
102 732
94 886
83 730
89 658
79 556
66 536
83 740
90 816
97 756
80 742
76 676
116 952
86 768
64 468
89 668
73 724
D =
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
>> [iter, optCond, time, w, gamma] = lsvm(A,d,nu,tol,maxIter,alpha, perturb,normalize)
;
Running time (CPU secs) = 0
Number of iterations = 70
Training accuracy = 0.83
>> w
w =
0.0993
-0.0140
>> x=[1:140];
>> y=(0.0993.*x - gamma)./0.0140;
>> plot(X(1:50),Y(1:50),'ob')
>> hold on
>> plot(X(51:100),Y(51:100),'or')
>> plot(x,y,'k')