Hadamard Extensions and the Identification of Mixtures of Product Distributions
Hadamard Extensions and the Identification of Mixtures of Product Distributions
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DOI：
10.1109/tit.2022.3146630
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发表时间：
2021-01
影响因子：
2.5
通讯作者：
Spencer Gordon;L. Schulman
中科院分区：
文献类型：
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作者：
Spencer Gordon;L. Schulman
$ n \ times k $矩阵M的Hadamard扩展$ \ Mathbb H（{\ Mathrm {M}}）$是其行的所有Hadamard产品的集合。这种结构对于$ n $二进制随机变量的$ K $产品分布的混合物的源识别（参数估计）至关重要。这种标识的必要要求是$ \ mathbb h（{\ mathrm {m}}）$具有完整的列等级；相反，如果除了每行都存在两个不相交的M，则可以识别M，每个行的扩展名都具有完整的列等级。因此，有必要理解$ \ MATHBB H（{\ MATHRM {M}}）$具有完整的列等级；我们在这个方向上提供两个结果。首先是，如果$ \ mathbb h（{\ mathrm {m}}）$具有完整的列等级，则最多有一组$ k-1 $ rows的m，其扩展已经具有完整的列等级。第二个是对M行中值的霍尔类型条件，足以确保全列等级为$ \ Mathbb H（{\ Mathrm {M}}）$。
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 $\mathbb H({\mathrm {m}})$ .