The technique deserves to be more well known than it is, because it gives good learning performance (when the associated kernel is a good choice for the problem), it is straightforward to implement, and it is very fast. I'm hoping that I can increase awareness by providing a simple implementation on a well-known dataset, and in that spirit here is a Matlab script which applies the technique to mnist. Before running this you need to download mnist in matlab format, and download maxent and lbfgs for matlab.

rand('seed',867); randn('seed',5309); tic fprintf('loading mnist'); % get mnist from http://cs.nyu.edu/~roweis/data/mnist_all.mat load('mnist_all.mat'); trainx=single([train0; train1; train2; train3; train4; train5; train6; train7; train8; train9])/255.0; testx=single([test0; test1; test2; test3; test4; test5; test6; test7; test8; test9])/255.0; st=[size(train0,1); size(train1,1); size(train2,1); size(train3,1); size(train4,1); size(train5,1); size(train6,1); size(train7,1); size(train8,1); size(train9,1)]; ss=[size(test0,1); size(test1,1); size(test2,1); size(test3,1); size(test4,1); size(test5,1); size(test6,1); size(test7,1); size(test8,1); size(test9,1)]; paren = @(x, varargin) x(varargin{:}); yt=[]; for i=1:10; yt=[yt; repmat(paren(eye(10),i,:),st(i),1)]; end ys=[]; for i=1:10; ys=[ys; repmat(paren(eye(10),i,:),ss(i),1)]; end clear i st ss clear train0 train1 train2 train3 train4 train5 train6 train7 train8 train9 clear test0 test1 test2 test3 test4 test5 test6 test7 test8 test9 fprintf(' finished: '); toc tic fprintf('computing random feature map'); % (uncentered) pca to 50 ... makes subsequent operations faster, % but also makes the random projection more efficient by focusing on % where the data is opts.isreal = true; [v,~]=eigs(double(trainx'*trainx),50,'LM',opts); trainx=trainx*v; testx=testx*v; clear v opts; % estimate kernel bandwidth using the "median trick" % this is a standard Gaussian kernel technique [n,k]=size(yt); [m,p]=size(testx); sz=3000; perm=randperm(n); sample=trainx(perm(1:sz),:); norms=sum(sample.^2,2); dist=norms*ones(1,sz)+ones(sz,1)*norms'-2*sample*sample'; scale=1/sqrt(median(dist(:))); clear sz perm sample norms dist; % here is the actual feature map: % Gaussian random matrix, uniform phase, and cosine d=4000; r=randn(p,d); b=2.0*pi*rand(1,d); trainx=cos(bsxfun(@plus,scale*trainx*r,b)); testx=cos(bsxfun(@plus,scale*testx*r,b)); fprintf(' finished: '); toc tic fprintf('starting logistic regression (this takes a while)\n'); % get @maxent and lbfgs.m from http://www.cs.grinnell.edu/~weinman/code/ % if you get an error about randint being undefined, change it to randi addpath recognition; addpath opt; addpath local; C0=maxent(k,d); [~,trainy]=max(yt'); options.MaxIter=300; options.Display='off'; C1=train(C0,trainy,trainx,'gauss',4.2813,[],[],[],options); % regularizer was chosen by cross-validation as follows %perm=randperm(n); %it=logical(zeros(1,n)); %it(perm(1:int32(0.8*n)))=1; %[C1,V]=cvtrain(C0,trainy(perm),trainx(perm,:),'gauss',10.^linspace(-4,4,20), ... % [],0,[],it,[],@accuracy); fprintf('finished: '); toc fprintf('train accuracy is %g\n',accuracy(C1,trainy,trainx)); [~,testy]=max(ys'); fprintf('test accuracy is %g\n',accuracy(C1,testy,testx));

Here's the result of running the script on my laptop:

>> clear all; cosplay loading mnist finished: Elapsed time is 2.227499 seconds. computing random feature map finished: Elapsed time is 6.994094 seconds. starting logistic regression (this takes a while) finished: Elapsed time is 219.007670 seconds. train accuracy is 0.99905 test accuracy is 0.9822This approaches the performance of the Gaussian kernel SVM, but with simplicity and speed. By trying different random feature maps, you can improve upon this result.

If you like this sort of thing, make sure to check out the Randomized Methods for Machine Learning workshop at NIPS 2013.

Thanks for publishing this code. Here's a slightly modified version for the Digit Recognizer competition at Kaggle:

ReplyDeletehttps://github.com/zygmuntz/kaggle-digits

It scores ~ 0.977.