Otherwise, here is my specific problem, I would also be interested in collaborating:
Is there any possibility to represent a probabilistic distribution on a set of exclusive hypotheses (Let us say a set of classes or system states: w_1, w_2,...., w_K such as
p(w_1)+p(w_2)+...+p(w_K)=1) by a single parameter (that preserves continuity)? This is possible via some parametric functions in the case of K=2. Is there any possibility for higher values of K?
Finding a continuous bijection between R and [0,1]^p would solve the problem.
Thanks for the answer Luis.
In 1911, L. E. J. Brouwer proved the result for any couple of (m,n) such that m!=n. Such bijections (between R^n and R^m) cannot be continuous (unfortunately :p ).
Would it be possible then to find a solution to my specific problem in some way? It's like to map a 3D color-space into a 1D one considering the color spaces continuous and preserving such mapping continuous. May be through some parametric functions? (The aim is to have 2 distributions are close to each other as long their parameters are close).
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seems to me you are looking for space-filling curves ; they are indeed continuous.
http://en.wikipedia.org/wiki/Space-filling_curve
the problem when using them for proximity query (which seems to be what you are aiming for) is that two points close on the curve are indeed close in the representation space (continuity ...) but the reverse is not true : due to the space filling behaviour, the curve "makes many turns" and two points close in representation space can be mapped very far away from each other (just look at the figures, it is obvious)
now, space filling curves have been used for proximity queries (find approximate nearest neighbours of a given point, for instance, of find if two points are close in representation space) but at the cost of using many different such space-filling curves simultaneously (see reference attached)
for a commonly used space-filling curve, see the Z-curve :
http://en.wikipedia.org/wiki/Z-order_%28curve%29
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In short, there are bijections between [0,1] and [0,1]^n (n>1) but such bijections cannot be continuous. See detailed post below.
@Etienne But I believe such bijections are not continuous! If you think the answer is yes, could you give me an example please?
There are bijections between I and I^n = I x I x ... x I (n > 1 factors), where I denotes the closed interval [0,1] (see "space filling curves").
However, there are no continuous bijections between I and I^n (n>1).
Suppose a continuous bijection between I and I^n existed for some n>1. Then:
- Such a function would be a homeomorphism (since I is compact and I^n is Hausdorff). See for example Theorem 26.6 in "Topology, 2nd edition", J.R. Munkres..
- However, I and I^n are not homeomorphic (a consenquence of the invariance of domain theorem, see, for example, Munkres' book for details). See also the notion of dimension of a topological manifold.
Conclusion: a bijection between I and I^n (n>1) cannot be continuous.
@Majid Two points close in the space may be far in the curve!
@Majid: Peano curves (or space filling curves) are not injective.
Another way to convice yourself that [0,1] cannot be in continuous bijection with [0,1]^2:
- such a continuous bijection would be a homeomorphism (since [0,1] is compact and [0,1]^2 is Haussdorff);
- deleting a point from the interior of [0,1] yields a topological space with two connected components while deleting a point from the interior of [0,1]^2 yields a topological space with a single connected component.
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