Positive and Negative Definite Kernels on Trees

Abstract

Let X= (V, E) be a tree with the set V of vertices and the set E of edges. For any x ∈ V we will denote by N(x) the neighbourhood of x, i.e. the set v ∈ V: d(v, z) ≤ 1. Suppose that for any x ∈ V we have a fixed positive definite matrix A(x) = (a(v, x, w)) _{v, w∈N(x)} such that a(v, x, v) = 1 for any v ∈ N(x). We define the kernel φ: V × V → C in the following way: if x, y ∈ V and [x, y] = x ₀ = x, x ₁, x ₂,...,x _n = y ⊂ V is the geodesic from x to y, then we put

$$\varphi (x,y) = \prod\limits_{i = 0}^n {a(x_{i - 1} ,x_i ,x_{i + 1} ),} $$

where, by definition, x ₋₁ = x ₀ = x and x _n +1 = x _n = y. In particular φ(x, x) = 1.

$$\beta (x,y) = \prod\limits_{i = 0}^{n - 1} {a(x_{i - 1} ,x_i ,x_{i + 1} ),} $$

α(x,x) = 1, which will help us in computations. Note that for any i ∈ {0,1,2,..., n} we have

$$\varphi (x,y) = \beta (x,x_i )a(x_{i - 1} ,x_i ,x_{i + 1} )$$

(1)