is "positive definite" in context of kernels and inner products the same thing?

Question

I am trying to understand kernels and their relation to feature maps, and particularly the positive definite requirement.

From wikipedia's article on positive definite kernels, let $X$ be a nonempty set. Then, the symmetric function $K:X\times X\to \mathbb{R}$ is called a positive definite (p.d.) kernel on $X$ if

$$\sum_{i,j=1}^n c_i c_j K(x_i,x_j) \geq 0$$

holds for any $n \in \mathbb{N}$, $x_1,...,x_n \in X$, $c_1,...,c_n \in \mathbb{R}$.

We can also use a feature map to define a kernel. Let $F$ be a Hilbert space, and $(\cdot,\cdot)_F$ the corresponding inner product. Any map $\Phi: X \to F$ is called a feature map, and we can define a kernel via

$$K(x,y) = (\Phi(x), \Phi(y))_F$$

But the RHS, being an inner product, must satisfy $(\Phi(x), \Phi(x))_F \geq 0$, with equality only when $\Phi(x) = \mathbf{0}$.

Are these two definitions for positive definiteness logically equivalent? If not, is one more general than the other? From the above formulation, I can't quite see how to get to one from the other... Additionally, wikipedia says that "every p.d. kernel, and its corresponding RKHS, have many associated feature maps", which makes me think the two are not equivalent.

source: https://en.wikipedia.org/wiki/Positive-definite_kernel

Answer 1

A symmetric function on a set defines a symmetric bilinear form on the vector space of functions with finite support, and it is positive definite iff this bilinear form is positive definite.

One way to get candidates for such functions is to write down feature maps. The symmetric functions defined via feature maps always have the property that the corrsponding bilinear map is positive semidefinite, but more has to be done to check that they're positive definite.