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In machine learning, a hyperplane is a fundamental concept used in various algorithms, especially in the linear classification and regression. 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. It is a geometric entity in a multidimensional space that is one dimension less than the space itself. In a two-dimensional space (a plane), a hyperplane is a straight line. In a three-dimensional space, it's a flat two-dimensional plane, and in higher dimensions, it's a flat, (n-1)-dimensional subspace in an n-dimensional space.

The key use of hyperplanes in machine learning is to separate data points into different classes or predict numerical values based on their positions relative to the hyperplane. Here are some common applications:

1. Linear Classification: In binary classification, a hyperplane can be used to separate data points belonging to two different classes. The data points on one side of the hyperplane are assigned to one class, while those on the other side are assigned to the other class. Common algorithms that use hyperplanes for classification include Support Vector Machines (SVM) and linear logistic regression.

2. Regression: In linear regression, a hyperplane is used to model the relationship between input features and a continuous target variable. The goal is to find the hyperplane that best fits the data points, minimizing the error between the predicted values and the actual target values.

3. Dimensionality Reduction: Principal Component Analysis (PCA) is a dimensionality reduction technique that involves finding hyperplanes in high-dimensional data space that capture the most variance. These hyperplanes, also known as principal components, are used to reduce the dimensionality of the data while retaining as much information as possible.

In classification, a key concept is the margin, which is the perpendicular distance from a data point to the hyperplane. Support Vector Machines (SVM), for example, aim to find the hyperplane that maximizes the margin between the classes, which helps create a robust decision boundary.

Hyperplanes are crucial in linear models, and they serve as the foundation for more complex algorithms and techniques used in machine learning, providing a way to generalize relationships between input features and output predictions.

Figure 3855a shows linear regression plotted with hyperplane η = θ^T x. For Gaussian distribution, η is equal to μ. (see page3868) The Gaussian distributions are along the hyperplane.

Figure 3855a. Linear regression plotted with hyperplane θ^T x (Python code).

Figure 3855b shows the linear learning model interaction with input and distribution. During learning process, a model learns parameters like θ through the learning process but the ditribution is not learnt. These parameters capture the relationships between input features and the target variable. the distribution of the data, which represents the underlying statistical properties of the dataset, is typically not learned explicitly in many machine learning models. Instead, the model makes certain assumptions about the distribution (e.g., assuming a normal distribution) but doesn't directly estimate the entire distribution. This separation of learning parameters and modeling the data distribution is a common practice in various machine learning algorithms.

Figure 3855b. Linear learning model.

The decision boundaries in Figure 3855c represent hyperplanes. In a real Softmax Regression model, these boundaries would be learned from the data.

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