A discriminant function
is a mathematical function that is used to classify or discriminate between different classes or categories. It takes in input variables and assigns them to specific classes based on their values. The discriminant function calculates a score or decision boundary that determines the class membership of the input.
Support Vector Machines
(SVM) is a popular and powerful machine learning algorithm that uses discriminant functions for classification and regression tasks. SVM aims to find an optimal hyperplane in a high-dimensional feature space that separates different classes or regression targets with the largest margin or distance between them. The hyperplane is defined by a subset of training data points called support vectors. In SVM, the choice of the discriminant function depends on the type of problem being solved. For binary classification, SVM uses a linear discriminant function that separates the classes with a linear decision boundary. However, SVM can also utilize nonlinear discriminant functions by applying kernel tricks, which transform the input space into a higher-dimensional feature space, allowing for nonlinear separation of classes.
The SVM algorithm involves finding the optimal hyperplane by solving a convex optimization problem. It aims to maximize the margin between the support vectors of different classes while minimizing classification errors. SVMs have the advantage of being effective in high-dimensional spaces, handling complex data patterns, and being less prone to overfitting. SVMs have a wide range of applications, including image classification, text classification, bioinformatics, and finance. They are known for their ability to handle both linearly separable and nonlinearly separable data and have been widely studied and used in the machine learning community.