Is there any example of such a function $f$, preferably one defined on all $V^n$ and all positive integers $n$ where $V$ is some vector space? It must satisfy the following:
- $f(T) = f(\sigma(T))$ for any $T \in V^n$ and permutation $\sigma$
- $f((X,t,t)) = f((X,t))$ for any (possibly empty) sequence $X$ and $t \in V$
- $f(a+T) = a+f(T)$ for any $T \in V^n$ and $a \in V$ where $a+(t,...,u) = (a+t,...,a+u)$
- $f(B*T) = B*f(T)$ for any $T \in V^n$ and diagonal matrix $B$ where $B*(t,...,u) = (B*t,...,B*u)$
- $f$ is continuous on $V^n$ for each positive integer $n$
If there is no closed form, an algorithm to approximate it will be good enough.
The reason I am looking for such a function is that I want to find a way to determine the most probable original vector given a set of samples where some samples could be dependent on others. In another field it is called textual criticism but the criteria used there are extremely subjective, whereas I believe this model uses objective criteria and the results are reproducible, if this function exists in the first place. Permutation invariance and duplicate invariance are necessary to exclude identical copies (like those of internet-myths). Translation and stretch invariance seem to be logical requirements as well. If $f$ does not exist, can a proof be shown? Thanks a lot!