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SVM algorithms categorize binary data, with the goal of fitting the training set data in a way that minimizes the average of the hinge-loss function and L2 norm of the learned weights. This strategy avoids overfitting via Tikhonov regularization and in the L2 norm sense and also corresponds to minimizing the bias and variance of our estimator ...
The first term is the objective function from ordinary least squares (OLS) regression, corresponding to the residual sum of squares. The second term is a regularization term, not present in OLS, which penalizes large values. As a smooth finite dimensional problem is considered and it is possible to apply standard calculus tools.
Given the binary nature of classification, a natural selection for a loss function (assuming equal cost for false positives and false negatives) would be the 0-1 loss function (0–1 indicator function), which takes the value of 0 if the predicted classification equals that of the true class or a 1 if the predicted classification does not match ...
When learning a linear function , characterized by an unknown vector such that () =, one can add the -norm of the vector to the loss expression in order to prefer solutions with smaller norms. Tikhonov regularization is one of the most common forms.
In many applications, objective functions, including loss functions as a particular case, are determined by the problem formulation. In other situations, the decision maker’s preference must be elicited and represented by a scalar-valued function (called also utility function) in a form suitable for optimization — the problem that Ragnar Frisch has highlighted in his Nobel Prize lecture. [4]
Suppose a vector norm ‖ ‖ on and a vector norm ‖ ‖ on are given. Any matrix A induces a linear operator from to with respect to the standard basis, and one defines the corresponding induced norm or operator norm or subordinate norm on the space of all matrices as follows: ‖ ‖, = {‖ ‖: ‖ ‖ =} = {‖ ‖ ‖ ‖:} . where denotes the supremum.
There are a number of matrix norms that act on the singular values of the matrix. Frequently used examples include the Schatten p-norms, with p = 1 or 2. For example, matrix regularization with a Schatten 1-norm, also called the nuclear norm, can be used to enforce sparsity in the spectrum of a matrix.
It was proven in 2014 that the elastic net can be reduced to the linear support vector machine. [7] A similar reduction was previously proven for the LASSO in 2014. [8] The authors showed that for every instance of the elastic net, an artificial binary classification problem can be constructed such that the hyper-plane solution of a linear support vector machine (SVM) is identical to the ...