Seonglae ChoGeneralized Distance or Vector Similarity
The positive-definite kernel, denoted as , calculates the inner product of feature-mapped inputs. The notation <> represents Vector Similarity, where is the vector kernel mapping function:
Core Properties:
- Symmetry - A valid kernel function must be symmetric
- Semi-definiteness - The kernel matrix must be positive semi-definite
Mathematical Conditions:
- Symmetry: (Commutative Property)
- Positive Semi-definiteness: for any
Key Features and Applications:
- Enables classification of nonlinear data using linear classifiers through higher-dimensional mapping
- Functions as a Feature Map that computes inner products between transformed input data
- Maintains nonlinear characteristics due to the rarity of orthogonal relationships in high-dimensional spaces
- Generates additional dimensions using existing dimensional data
- Serves as similarity metrics, particularly in the case of Gaussian Kernel
Theory
Positive-definite kernel
In operator theory, a branch of mathematics, a positive-definite kernel is a generalization of a positive-definite function or a positive-definite matrix. It was first introduced by James Mercer in the early 20th century, in the context of solving integral operator equations. Since then, positive-definite functions and their various analogues and generalizations have arisen in diverse parts of mathematics. They occur naturally in Fourier analysis, probability theory, operator theory, complex function-theory, moment problems, integral equations, boundary-value problems for partial differential equations, machine learning, embedding problem, information theory, and other areas.
https://en.wikipedia.org/wiki/Positive-definite_kernel