Feature space interpretation of SVMs with indefinite kernels

Area of Science:

• Machine Learning
• Computational Mathematics
• Data Science

Background:

• Support Vector Machines (SVMs) are widely used for classification tasks.
• Understanding SVMs is well-established for conditionally positive definite (cpd) kernels.
• Non-cpd (indefinite) kernels are increasingly used in practice, yielding empirical success but lacking theoretical interpretation.

Purpose of the Study:

• To provide a geometric interpretation for SVM classifiers utilizing indefinite kernel functions.
• To establish a theoretical framework and motivation for the application of indefinite SVMs.
• To lay the groundwork for further theoretical analysis and practical guidelines for indefinite SVMs.

Main Methods:

• Geometric interpretation of SVMs with indefinite kernels.
• Demonstration that these SVMs act as optimal hyperplane classifiers.
• Analysis based on the minimization of distances between convex hulls in pseudo-Euclidean spaces.

Main Results:

• SVMs with indefinite kernels are shown to be optimal hyperplane classifiers.
• Classification is achieved by minimizing distances between convex hulls, not by margin maximization.
• A sound theoretical framework and motivation for indefinite SVMs are established.

Conclusions:

• The geometric interpretation provides a basis for understanding indefinite SVMs.
• This framework supports further theoretical investigations, such as uniqueness analysis.
• The findings can lead to practical guidelines for assessing the suitability of indefinite SVMs.