I have been thinking about this problem and can’t come up with a solution for it. Perhaps someone here with a better quill may be able to!
Given a set $S=\{s_1,s_2,...,s_N\}$ of $N$ real numbers, $s_i\in\mathbb{R} \ \forall i$, find the minimum set $A=\{a_1,a_2,...,a_M\}$ of $M$ real numbers such that the sum of some combination of distinct elements of $A$ i.e. ($a_1+a_3$) or ($a_4+a_{35}+a_{132}$) etc, yields any element of $S$. Evidently $M=N$ must be an upper bound as it corresponds to a one-to-one mapping of the elements of $A$ to $S$, i.e. $a_i=s_i \forall i$.
For example, if $S=\{1,2,3\}$ clearly the minimum $M=2$ with $A=\{1,2\}$ since $s_1=a_2, s_2=a_2$ and $s_3=a_1+a_2$.
I’m not a mathematician but I guess this could be expressed as something like $$ \min M :s_i=\sum_n^{M}a_n\pi(n,i) \quad \forall i $$ where $\pi(n,i)\in\{0,1\}\forall n,i$ is a function you may choose depending on your needs.
I fear nothing can be said and that the answer depends on $S$.