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Hello, I will explain how SVM algorithm works. This video will explain the support vectormachine for linearly separable binary sets Suppose we have this two features, x1 andx2 here and we want to classify all this elements You can see that we have the class squareand the class rectangle So the goal of the SVM is to design a hyperplane,here we define this green line as the hyperplane, that classifies all the training vectors intwo classes Here we show two different hyperplanes whichcan classify correctly all the instances in this feature setBut the best choice will be the hyperplane that leaves the maximum margin from both classesThe margin is this distance between the hyperplane and the closest elements from this hyperplaneWe have the case of the red hyperplane we have this distance, so this is the margin,which we represent by z1 And in the case of the green hyperplane wehave the margin that we call z2 We can clearly see that the value of z2 isgreater than z1 So the margin is higher in the case of thegreen hyperplane, so in this case the best choice will be the green hyperplaneSuppose we have this hyperplane, this hyperplane is defined by one equation, we can state thisequation as this one We have a vector of weights plus omega 0 andthis equation will deliver values greater than 1 for all the input vectors which belongsto the class 1, in this case the circles And also, we scale this hyperplane so thatit will deliver values smaller than -1 for all values which belongs to class number 2,the rectangles We can say that this
distance to the closestelements will be at least 1, the modulus is 1From the geometry we know that the distance between a point and a hyperplane is computedby this equation So the total margin which is composed by thisdistance will be computed by this equation And the aim is that minimizing this term willmaximize the separability When we minimize this weight vector we willhave the biggest margin here that will split this two classesTo minimize this weight vector is a nonlinear optimization task, which can be solved bythis conditions (KKT), which uses Langrange multipliersThe main equations state that the value of omega will be the solution of this sum hereAnd we also have this other rule. So when we solve these equations, trying to minimizethis omega vector, we will maximize the margin between the two classes which will maximizethe separability the two classes Here we show a simple exampleSuppose we have these 2 features, x1 and x2, and we have these 3 valuesWe want to design, or to find the best hyperplane that will divide this 2 classesSo we know that we can see clearly from this graph that the best division line will bea parallel line to the line that connects these 2 values hereSo we can define this weight vector, which is this point minus this other point. So wehave the constant a and 2 times this constant aNow we can solve this weight vector and create the hyperplane equations considering thisweight vector We must discover the values of this a hereSince we have this weight vector omega here, we can substitute the values of this pointand also using
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