Symmetric measures via moments
نویسنده
چکیده
The uniqueness part of the problem of moments is concerned with whether a (multivariate) measure with finite (mixed) moments is uniquely determined by its moments. This work generalizes the above question by considering families of measures that are invariant under finite groups of (nonsingular) linear transformations. Uniqueness is then considered relative to a given family of invariant measures, and the totality of mixed moments is then replaced by the corresponding invariant polynomials. It is further shown how various sufficient conditions for (ordinary) determinacy, such as, for example, the extended (multivariate) Carlman condition, can be adapted to the new context via generators of the algebra of the invariant polynomials; that the number of such generators is finite, is known from the theory of polynomial invariants of finite groups. Several associated computational issues are discussed with a view toward model selection in the presence of such symmetries. A distribution of minuscule subimages extracted from a large database of natural images, along with generators for the relevant invariances, is discussed to illustrate the above concepts.
منابع مشابه
un 2 00 4 Symmetric Measures via Moments
A finite G ≤ GL(m, R) fixes Ω ⊂ R and induces its action on P , the set of probability distributions on Ω. P is the set of distributions invariant under this action. We consider models based on P. Ignoring the invariance, a common approach to modeling P ∈ P is to progressively match its moments. Among all distributions with a requested match, one reasonable choice is P ′ that maximizes the entr...
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