title: Confusion related to kernel svm
body: <p>I have this confusion related to kernel svm. I read that with kernel svm the number of support vectors retained is large. That's why it is difficult to train and is time consuming. I didn't get this part why is it difficult to optimize. Ok I can say that noisy data requires large number of support vectors. But what does it have to do with the training time.</p>
  
  <p>Also I was reading another <a href="https://www.google.com/url?sa=t&amp;rct=j&amp;q=&amp;esrc=s&amp;source=web&amp;cd=1&amp;cad=rja&amp;ved=0CDUQFjAA&amp;url=http://jmlr.csail.mit.edu/proceedings/papers/v22/zhang12d/zhang12d.pdf&amp;ei=90dpUcTeJcXu0gHylYHoDQ&amp;usg=AFQjCNE4mUPPsSiyQBW5FY_IyeRT7Im9Og&amp;sig2=r5oZey9nhHIW_MOZf6lGfg&amp;bvm=bv.45175338,d.dmQ" rel="nofollow">article</a> where they were trying to convert non linear SVM kernel to linear SVM kernel. In the case of linear kernel it is just the dot product of the original features themselves. But in the case of non linear one it is RBF and others. I didn't get what they mean by "manipulating the kernel matrix imposes significant computational bottle neck". As far as I know, the kernel matrix is static isn't it. For linear kernel it is just the dot product of the original features. In the case of RBF it is using the gaussian kernel. So I just need to calculate it once, then I am done isn't it. So what's the point of manipulating and the bottleneck thinkg</p>
  
  <p><strong>Support Vector Machine (SVM) (Cortes and Vap- nik, 1995) as the state-of-the-art classification algo- rithm has been widely applied in various scientific do- mains. The use of kernels allows the input samples to be mapped to a Reproducing Kernel Hilbert S- pace (RKHS), which is crucial to solving linearly non- separable problems. While kernel SVMs deliver the state-of-the-art results, the need to manipulate the k- ernel matrix imposes significant computational bottle- neck, making it difficult to scale up on large data.</strong></p>