Reconstructing Multivariate Observations with Possible Multiplicities from Sample Statistics

Algebraic methodologies are developed for addressing the inverse problem of recovering a collection of bivariate observations from selected subsets of their marginal and joint integer moments. Particular attention is devoted to the case of samples containing duplicates. The multivariate analogues emerge from suitable generalizations of the bivariate constructions. We also examine the problem of reconstructing a sample from its marginal moments together with its component-wise ranks. Extensions to complex-valued data are also considered. The results established in this paper demonstrate that an appropriately chosen family of marginal or joint moments can, in fact, encode the entirety of the information contained in the original data set, in the sense that the latter can be recovered uniquely from the former. The proposed methodologies are essentially mathematical, with statistical samples representing one of several possible applications. A series of illustrative examples highlights the scope of the methods.