Hadamard Extensions and the Identification of Mixtures of Product Distributions

The Hadamard Extension $\mathbb H({\mathrm {m}})$ of an $n \times k$ 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 $\mathbb H({\mathrm {m}})$ 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 $\mathbb H({\mathrm {m}})$ has full column rank; we provide two results in this direction. The first is that if $\mathbb H({\mathrm {m}})$ 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 $\mathbb H({\mathrm {m}})$ .