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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:

  1. $f(T) = f(\sigma(T))$ for any $T \in V^n$ and permutation $\sigma$
  2. $f((X,t,t)) = f((X,t))$ for any (possibly empty) sequence $X$ and $t \in V$
  3. $f(a+T) = a+f(T)$ for any $T \in V^n$ and $a \in V$ where $a+(t,...,u) = (a+t,...,a+u)$
  4. $f(B*T) = B*f(T)$ for any $T \in V^n$ and diagonal matrix $B$ where $B*(t,...,u) = (B*t,...,B*u)$
  5. $f$ 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!

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    I don't understand condition 2, as I don't see what elements of $V^n$ are $(X,t,t)$ and $(x,t)$. Can you elaborate on this please?2011-12-27
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    @Alex: He probably means a function on the set of _all_ finite sequences of vectors, rather than a function on $V^n$.2011-12-27
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    @Zhen: Ah, that makes sense. User213820: If this is the case, you should edit your question to replace $V^n$ with $\bigoplus\limits_{n=1}^\infty V$.2011-12-27
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    I still think condition 2 needs clarification. If OP explains what he meant by this notation it would be best for all of us.2011-12-27
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    Discard duplicates and take the mean? That satisfies all your criteria.2011-12-27
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    @Alex: Yes I meant what Zhen said. Sorry I did not know there was such a standard notation so I had put "for any n".2011-12-27
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    For condition 2, an example would be $f((x,y,t,t))=f((x,y,t))$2011-12-27
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    @Rahul: At first I wanted continuity but removed it.. Sorry I should have left it in.. I will edit my question to reflect that as well as Zhen's clarification.2011-12-27
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    @Patrick, I think user21820 is using the expression $f((X,t,t))$ to denote the function $f$ applied to the concatenation of two sequences $X$ and $(t,t)$.2011-12-27
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    @Rahul : OP made it clear in his comments, but thanks. =)2011-12-27
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    The part with the matrix is weird though : You are only saying that $V$ is a vector space, are you also assuming it to be $K^n$ for some field $K$? Because you're writing in condition 4 that there's a matrix $B$ that's diagonal over some basis of $V$.2011-12-27
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    @Patrick: Not quite; I simply wanted stretch-invariance but do not know how best to state it. It is of course preferable if $V$ is $F^r$ for some field $F$, but if matrix multiplication is defined in terms of an action of a field on the elements of V, it would suffice for me.2011-12-28

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