Positive Definite Kernels

see also:

• Positive Definiteness

do kernels exhibit the property that K<x,x’>=K<x’,x> ?

Yes, many kernels do exhibit the property that (K(x, x’) = K(x’, x)), which is known as the symmetry property of kernel functions. This symmetry is a characteristic of positive definite kernels, which are commonly used in machine learning, statistics, and other areas of computational mathematics, especially within the framework of Reproducing Kernel Hilbert Spaces (RKHS).

Positive Definite Kernels

A kernel (K: X \times X \rightarrow \mathbb{R}) (or (\mathbb{C}) for complex-valued functions) on a non-empty set (X) is called positive definite if it satisfies two main conditions:

1. Symmetry: For all (x, x’ \in X), (K(x, x’) = K(x’, x)). This means that the kernel’s value is the same regardless of the order of its arguments, reflecting the notion of an inner product’s symmetry in vector spaces.

2. Positive Definiteness: For any finite set of points (x_1, x_2, \ldots, x_n \in X) and any set of real (or complex) numbers (c_1, c_2, \ldots, c_n), the sum

   $$\sum _{i=1}^n\sum _{j=1}^n{c}_i{c}_{j}K(x_i,x_j)\ge 0$$

   is always non-negative. This condition ensures that the kernel behaves similarly to the dot product in an inner product space, measuring similarity in a generalized feature space.

Examples of Symmetric Kernels

Many widely used kernels possess this symmetry property, such as:

• Linear Kernel: (K(x, x’) = x \cdot x’), where the dot product is inherently symmetric.

• Polynomial Kernel: (K(x, x’) = (1 + x \cdot x’)^d), for some positive integer (d), also exhibits symmetry due to the symmetry of the dot product.

• Radial Basis Function (RBF) or Gaussian Kernel: (K(x, x’) = \exp(-\gamma |x - x’|^2)), where (\gamma) is a positive constant, and the Euclidean distance squared (|x - x’|^2) is symmetric in its arguments.

Importance of Symmetry

The symmetry property of kernels is crucial for their use in kernel methods, such as in support vector machines (SVMs) and in the construction of kernel matrices (Gram matrices) used in various machine learning algorithms. The symmetry ensures that the kernel matrix is symmetric, which is a desirable property for mathematical and computational reasons, including the spectral decomposition of matrices.

In summary, the symmetry of kernels, (K(x, x’) = K(x’, x)), is a fundamental property that aligns with the concept of similarity measurement and the construction of Hilbert Space of functions where these kernels are employed. It facilitates the use of kernel methods in efficiently solving linear and nonlinear problems by operating in implicitly defined high-dimensional feature spaces.