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The optimal margin classifier is the hyperplane that correctly classifies all data while being farthest away from the data points. The optimal margin classifier is the maximal margin classifier, which is defined in the rare case where two classes are linearly separable. The maximal margin classifier is the best hyperplane to correctly classify new data and is particularly in the field of support vector machines (SVMs). However, this SVM requires the two classes to be completely linearly separated.

The terms "Optimal Margin Classifier" and "Maximum Margin Separator" are often used interchangeably and refer to the same concept in SVMs. In SVMs, the goal is to find a hyperplane that maximizes the margin between different classes of data points. This hyperplane is sometimes referred to as the "maximum margin separator" or "optimal margin classifier" because it achieves the maximum separation between classes by having the largest possible margin. 

The margin is defined as the distance between the hyperplane and the nearest data points from each class. The idea behind this is to create a classifier that not only separates the data but also does so with a significant "safety margin," making it more robust and less prone to overfitting.

Key characteristics of the optimal margin classifier:

1. Maximized Margin: The hyperplane is positioned to maximize the distance between the data points of different classes. This distance is the margin.

2. Support Vectors: The data points that are closest to the hyperplane and define the margin are called support vectors. They play a crucial role in determining the position and orientation of the hyperplane.

3. Margin is Optimized: The optimal margin classifier seeks to find the hyperplane that not only separates the data but also maximizes the margin. It effectively "balances" the separation and generalization.

4. Robustness: By maximizing the margin, the classifier becomes more robust and less sensitive to noise in the data.

Mathematically, the goal of finding the optimal margin classifier can be expressed as an optimization problem. The hyperplane is defined by a weight vector and a bias term, and the objective is to find the values that maximize the margin while correctly classifying the data points.

If the data is not linearly separable (i.e., a single hyperplane cannot perfectly separate the classes), SVMs can use techniques like the kernel trick to transform the data into a higher-dimensional space where it may become linearly separable.

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