title: Query about SVM mapping of input vector? And SVM optimization equation
body: <p>I have read through a lot of papers and understand the basic concept of a support vector machine at a very high level. You give it a training input vector which has a set of features and bases on how the "optimization function" evaluates this input vector lets call it x, (lets say we're talking about text classification), the text associated with the input vector x is classified into one of two pre-defined classes, this is only in the case of binary classification. </p>
  
  <p>So my <strong>first question</strong> is through this procedure described above, all the papers say first that this training input vector x is mapped to a higher (maybe infinite) dimensional space. So what does this mapping achieve or why is this required? Lets say the input vector x has 5 features so who decides which "higher dimension" x is going to be mapped to?</p>
  
  <p><strong>Second question</strong> is about the following optimization equation:</p>
  
  <p>min 1/2 wi(transpose)*wi + C Σi = 1..n ξi </p>
  
  <p>so I understand that w has something to do with the margins of the hyperplane from the support vectors in the graph and I know that C is some sort of a penalty but I dont' know what it is a penalty for. And also what is  ξi representing in this case.</p>
  
  <p>A simple explanation of the second question would be much appreciated as I have not had much luck understanding it by reading technical papers.</p>