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I have two questions about the behavior of the Gaussian Kernel matrix at small scales. I had asked a similar question in math.stackexchange but did not get any response. https://math.stackexchange.com/questions/2559201/behavior-of-the-determinant-of-the-gaussian-kernel-matrix-near-0

Let $x_1,x_2,\dots,x_p$ be distinct points in $\mathbb{R}^d$ in general position where $p \geq 2$. Let $K(\lambda) = ( \exp(-\lambda\|x_i - x_j\|^2) )_{1\leq i,j \leq p}$ denote the Gaussian kernel matrix for $x_i$'s with scale parameter $\lambda > 0$ and let $\Delta(\lambda) := \det K(\lambda)$.

For a general $p$:

  1. $\Delta(\lambda)$ is an analytic function of $\lambda$, and its first $p-2$ derivatives vanish. What is the smallest value of $k$ for which $\Delta^{(k)}(0) \neq 0?$ We can see $k=1$ when $p=2$.

  2. Is there a $t$ such that $\lim_{\lambda \to 0^+} \lambda^t (K(\lambda))^{-1}$ exists and is not zero? We can see $t=1$ when $p=2$.

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