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Module 3: Linear Classification and Advanced Methods

20267m

Explain Support Vector Machines. Discuss the concept of maximum-margin hyperplane, support vectors, and soft margin.

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Support Vector Machine (SVM)

SVM finds a separating hyperplane with the largest margin between classes. The closest training examples are support vectors; they determine the boundary.

For a linear separator (w^Tx+b=0), the geometric margin is proportional to (1/\lVert w\rVert). A soft-margin SVM allows violations using slack variables (\xi_i):

[\min_{w,b,\xi};\frac12\lVert w\rVert^2+C\sum_i\xi_i]

subject to (y_i(w^Tx_i+b)\ge1-\xi_i), (\xi_i\ge0). Large (C) penalizes violations more strongly; smaller (C) allows a wider margin with more violations. Kernel functions can create nonlinear decision boundaries by implicitly mapping data into a higher-dimensional feature space.

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