Is there a bijection between transformations (on vector spaces) and kernel functions?

Question

I was reading about kernel functions (the functions they use in support vector machines to transform the data points into a different space, called the kernel trick or something). In mathematical terms, instead of computing $\rho(x)^T\rho(y)$ for some vectors $x,y\in \mathbb{R}^n$ and some transformation $\rho: \mathbb{R}^n \rightarrow \mathbb{R}^m$, we can find some function $K(x,y)=\rho(x)^T\rho(y)$ where we don't have to explicitly compute $\rho(x),\rho(y)$. Is there a bijection between $K$ and $\rho$? I'm not sure if "not explicitly computing" is really well defined, and I'm not exactly sure how to prove this and haven't come up with any counterexamples (but maybe I'm just missing something easy).

Answer 1

No. Given $K$, there can be multiple $\rho$.

Suppose $K(x,y) = \rho(x)^T\rho(y)$

Consider $\rho_1(x) = U \rho(x)$ where $U$ is orthogonal, then $\rho_1(x)^T\rho_1(y)=\rho(x)^T U^TU \rho(y)=K(x,y)$.