Response 2/3
> Is the design of MAGAREA unique? Cannot we replace $\exp$ in the definition of $\zeta$ with another increasing and convex function?
The computation of MAGAREA is unique insofar as we require $\zeta(t)=f(-td)$ being invertible for all $t \in \mathbb{R}^+$, which, to the best of our understanding, won't necessarily hold for any increasing and convex function $f$. For example, take $f(x)=x^{1000}$, so that $\zeta_{ij}(t) = (t d_{ij})^{1000}$ which is increasing and convex for $td > 0$. Then, for some low values of $t$ close to zero, $\zeta$ will be singular and hence not invertible.
We know, however, that any positive definite matrix is invertible so that we can define the magnitude of any such matrix $\zeta$ [(Leinster 2017)](https://arxiv.org/pdf/1606.00095). Given this motivation, $\exp$ is a somewhat “canonical” choice for defining the magnitude of a metric space because it is a prime example of a strictly positive definite kernel, which ensures invertibility of the similarity matrix for any negative definite metric $d$ [(Feragen et al. 2015)](https://ieeexplore.ieee.org/document/7298922).
From a category-theoretic perspective, magnitude represents a generalised notion of size for a metric space, which is a special type of monoidal categories. The exponential kernel then has been chosen for defining the magnitude of a metric space because of its multiplicative properties i.e. it is necessary that $f(x+y)=f(x) \cdot f(y)$ [(Leinster 2021, p.212)](https://arxiv.org/pdf/2012.02113) to define a valid notion of size. This essentially forces, $f(x) = c^{-x}$ for some constant $c$. Hence, the choice of $e$ as a basis is arbitrary and any other positive constant could be used, which is equivalent to re-scaling the distances.
We also want to note that the current definition of magnitude has revealed useful theoretical and geometric insights, such as proving the connection between magnitude and the curvature of a metric space. Hence, if we choose any $f$ that does not fulfil the multiplicative property above, we lose the existing knowledge relating magnitude to geometrical properties of the underlying space, for example.
The standard formulation of $\zeta$ also has some appealing properties linking it to diversity. We have $\exp(-td_{ii})=\exp(0)=1$, so that the similarity of an observation to itself is always $1$. Further, we have $\lim_{t \to \infty} \exp(-t d_{ij}) = 0$, so that the similarity between distant points approaches zero asymptotically. Thus, $\zeta(t)$ gives a valid similarity matrix, whose entries are bounded by $[0,1]$, which is desirable for defining a (dis)similarity-based notion of diversity [(Leinster 2021, p.173)](https://arxiv.org/pdf/2012.02113). **In fact this behaviour is the practical reason why magnitude can be interpreted as the effective number of points.**
Hence, we know that other choices of $\zeta$ are possible and we believe it is worth investigating the generalisation of our methods to other choices of kernels in future work. Nevertheless, we find that, for now, $\exp$ provides a useful “default” choice for defining diversity based on the considerations detailed above.