For what values of $\beta \in \mathbb{R}$ is $t(x-x')=-||x-x'||^\beta$ a kernel? For what values of $\beta \in \mathbb{R}$ is $t(x-x')=-||x-x'||^\beta$ a kernel?
I know that kernels of type $t(x-x')$ where $t$ is function that inverts the dissimilarity $x-x'$ into a similarity measure proportional to the kernel. In this case, by means of adding a negative sign in front.
I have tried to plot several cases via Python and I observed that with $\beta>0$ I have sort of "concave" function, if you could call it like that:

With $\beta=0$ it is constant -1, and with $\beta<0$ I obtain a "convex" function with a division by zero at the origin. But with all that I do not answer to the question because I see plots, but I do not know how prove by which values of $\beta$ $t$ is a valid kernel. I would really appreciate your help!
 A: A related kernel that is valid is
$$
k(x, y)
= \lVert x \rVert^\beta + \lVert y \rVert^\beta - \lVert x - y \rVert^\beta
$$
for $0 < \beta \le 2$;
see Example 15 of Sejdinovic, Sriperumbudur, Gretton, and Fukumizu, Equivalence of distance-based and RKHS-based statistics in hypothesis testing, Annals of Statistics 2013.
You can also use
$$
\lVert x - z \rVert^\beta + \lVert y - z \rVert^\beta - \lVert x - y \rVert^\beta
$$
for any fixed $z$ if you'd like.
In general, if you want a quick "could this possibly be a kernel" numerical check, it's far more productive to construct a kernel matrix for some random inputs and check its eigenvalues: if any are negative, then it can't be a valid kernel. If you do this a few times and don't get any negatives, then maybe it is valid. (You might get some -1e-7 values out due to numerical error, but any actually negative values would disqualify it.)
A: This will never work! No matter what $\beta$ you choose.
Take three distinct points in any $\mathbb{R}^n$ and the determinant of your kernel matrix $(t(x_i,x_j))$ will be negative for every $\beta$.
Background: A function $t(\|x-y\|)$ defines a positive kernel on every $\mathbb{R}^n$, iff $t:[0,\infty[\rightarrow\mathbb{R}$ is "totally monotone"; totally monotone functions are necessarily nonnegative.
For proofs see Chapters 5 and 6 of Wendland, "Scattered Data Approximation".
