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Published June 2022 | public
Journal Article

Hadamard Extensions and the Identification of Mixtures of Product Distributions


The Hadamard Extension H(m) of an n×k matrix m is the collection of all Hadamard products of subsets of its rows. This construction is essential for source identification (parameter estimation) of a mixture of k product distributions over n binary random variables. A necessary requirement for such identification is that H(m) have full column rank; conversely, identification is possible if apart from each row there exist two disjoint sets of rows of m, each of whose extension has full column rank. It is necessary therefore to understand when H(m) has full column rank; we provide two results in this direction. The first is that if H(m) has full column rank then there exists a set of at most k−1 rows of m, whose extension already has full column rank. The second is a Hall-type condition on the values in the rows of m, that suffices to ensure full column rank of H(m).

Additional Information

© 2021 IEEE. Manuscript received February 14, 2021; revised October 13, 2021; accepted January 5, 2022. Date of publication January 26, 2022; date of current version May 20, 2022. This work was supported in part by NSF under Grant CCF-1909972. An earlier version of this paper was posted at the ArXiv:2101.11688. The authors would like to thank the anonymous referees for their careful review which substantially improved the manuscript.

Additional details

August 22, 2023
October 23, 2023