Supplementary Material on Hyperplanes in Machine Learning

In my work “Mathematical Foundations of Machine Learning”, pages 285-286, based on the Bernoulli distribution, the logistic function is derived, resulting in equation (5.3.16) on page 286:

Applying this equation to explore linear discriminant analysis, it represents a line in two-dimensional space, and in multi-dimensional space, it becomes a hyperplane.

1. Another Definition of Hyperplane

1.1 Algebraic Definition

For a plane in three-dimensional space, if extended to n-dimensional space, we have a system of linear equations:

The set of solutions to this equation (where x and y are real numbers) forms a hyperplane, which can be expressed in vector form as:

1.2 Geometric Definition

Let V be a subspace of R. The collection obtained by translating V from the origin is called an affine space, as shown in the figure below:

In R, a hyperplane is an affine space of dimension n-1, or in other words, aside from R itself, the hyperplane is the affine space with the highest dimension.

The above two definitions are equivalent.

References

[1]. Affine Transformation [DB/OL]. https://lqlab.readthedocs.io/en/latest/math4ML/linearalgebra/affine.html