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“We will now switch to a Lagrangian formulation of the problem. There are two reasons for doing this. The first is that the constraints (12) will be replaced by constraints on the Lagrange multipliers themselves, which will be much easier to handle. The second is that in this reformulation of the problem, the training data will only appear (in the actual training and test algorithms) in the form of dot products between vectors. This is a crucial property which will allow us to generalize the procedure to the nonlinear case (Section 4).