稀疏矩阵分解与图的刻画
Sparse Matrix Decompositions and Graph Characterizations
摘要：
当零（即，在正定矩阵$A$中的稀疏性）在其Cholesky分解中被保留，反之亦然，这是Paulsen等人在Journal of Functional Analysis（85，pp 151 -178）中提出的。特别是，他们证明，在$A$中的零的模式被保留在$A$的Cholesky分解，在$A$中的零的模式必须对应于一个弦（或可分解）图与特定类型的顶点排序。因此，这一结果产生的弦图的稀疏正定矩阵的特征。它也被证明是非常有用的概率和统计分析马尔可夫随机场的零正定相关矩阵的概念密切相关的随机独立。现在，考虑正定矩阵$A$和由$A = LDL^T$给出的它的Cholesky分解，其中$L$是具有单位对角元素的下三角矩阵，而$D$是具有正元素的对角矩阵。本文证明了零的一个充要条件（即，在正定矩阵$A$中保持在其相关联的Cholesky矩阵$L$中，并且另外还保持在Cholesky矩阵$L^{-1}$的逆中的稀疏性）在于，零的模式对应于与特定类型的顶点排序相关联的共弦或齐次图。我们继续提供这类图的第二个特征，在图中的集团对应的子矩阵的行列式。这些结果增加了越来越多的文献在该领域的稀疏矩阵分解，也被证明是关键成分的概率分析的一类重要的马尔可夫随机场。
英文摘要：
The question of when zeros (i.e., sparsity) in a positive definite matrix $A$ are preserved in its Cholesky decomposition, and vice versa, was addressed by Paulsen et al. in the Journal of Functional Analysis (85, pp151-178). In particular, they prove that for the pattern of zeros in $A$ to be retained in the Cholesky decomposition of $A$, the pattern of zeros in $A$ has to necessarily correspond to a chordal (or decomposable) graph associated with a specific type of vertex ordering. This result therefore yields a characterization of chordal graphs in terms of sparse positive definite matrices. It has also proved to be extremely useful in probabilistic and statistical analysis of Markov random fields where zeros in positive definite correlation matrices are intimately related to the notion of stochastic independence. Now, consider a positive definite matrix $A$ and its Cholesky decomposition given by $A = LDL^T$, where $L$ is lower triangular with unit diagonal entries, and $D$ a diagonal matrix with positive entries. In this paper, we prove that a necessary and sufficient condition for zeros (i.e., sparsity) in a positive definite matrix $A$ to be preserved in its associated Cholesky matrix $L$, \, and in addition also preserved in the inverse of the Cholesky matrix $L^{-1}$, is that the pattern of zeros corresponds to a co-chordal or homogeneous graph associated with a specific type of vertex ordering. We proceed to provide a second characterization of this class of graphs in terms of determinants of submatrices that correspond to cliques in the graph. These results add to the growing body of literature in the field of sparse matrix decompositions, and also prove to be critical ingredients in the probabilistic analysis of an important class of Markov random fields.
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