Hadamard Powers and the Identification of Mixtures of Products

The Hadamard Power of a matrix is the matrix consisting of all Hadamard products of subsets of its rows. We obtain several results concerning when a Hadamard Power has full column rank. This question in turn is central to the following problem: given a mixture of $k$ product distributions on a list of binary random variables $X_1,\ldots,X_n$, can the probability model be identified from the joint statistics of the $X_i$.