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
