Let $x \in \mathbb{R}^n$ and $k$ is RBF kernel

$$k(x, x') := \exp \left(-\frac{\|x-x'\|^2}{2\sigma^2}\right)$$

and let $\mathbf{K}$ be the following $n \times n$ covariance matrix

$$\mathbf{K} = \begin{bmatrix} 1 & k(x_1, x_2) & \dots & k(x_1, x_n)\\ k(x_2, x_1) & 1 & \dots & k(x_2, x_n)\\ \vdots & \vdots & \ddots & \vdots \\ k(x_n, x_1) & k(x_n, x_2) & \dots & 1\\ \end{bmatrix}$$

In practice, the sum of the entries of matrix $\mathbf{K}^{-1}$ is small. How can I find the minimum upper bound for it?

Specifically, when $\mathbf{1} = [1, ... , 1] \in \mathbb{R}^n$, I am looking for $M$ such that

$$\mathbf{1} \mathbf{K}^{-1} \mathbf{1}^{T} \leq M$$

**Note:** A similar question has been asked here, but unfortunately it is not answered. Moreover, another similar question has been asked here, but the value of the entries are not determined.

Thank you in advance for your help!

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