模拟协方差和浓度图矩阵的部分正交化方法
A partial orthogonalization method for simulating covariance and concentration graph matrices
摘要：
协方差图和浓度图的结构学习方法通常在合成模型上进行验证，通常通过随机生成：（i）无向图和（ii）兼容的对称正定（SPD）矩阵来获得。为了确保（ii）中的正定性，通常会施加一个优势对角线。然而，由SPD矩阵中的非对角条目确定的所得到的图形模型中的链路强度在许多场景中非常弱。因此，恢复无向图的结构成为一个挑战，算法验证受到显着影响。在本文中，我们提出了一种替代方法，克服了这样的问题，但产生一个兼容的SPD矩阵。我们生成一个部分行正交矩阵因子，其中成对正交行对应于无向图中缺失的边。在数值实验中，从适度密集到稀疏的情况下，我们得到，随着维数的增加，我们模拟的链接强度是稳定的结构稀疏。重要的是，我们在一个真实的验证设置中展示了当使用我们提出的方法时，所有学习算法的结构恢复是如何大大改善的，从而产生了一个更现实的比较框架。
英文摘要：
Structure learning methods for covariance and concentration graphs are often validated on synthetic models, usually obtained by randomly generating: (i) an undirected graph, and (ii) a compatible symmetric positive definite (SPD) matrix. In order to ensure positive definiteness in (ii), a dominant diagonal is usually imposed. However, the link strengths in the resulting graphical model, determined by off-diagonal entries in the SPD matrix, are in many scenarios extremely weak. Recovering the structure of the undirected graph thus becomes a challenge, and algorithm validation is notably affected. In this paper, we propose an alternative method which overcomes such problem yet yielding a compatible SPD matrix. We generate a partially row-wise-orthogonal matrix factor, where pairwise orthogonal rows correspond to missing edges in the undirected graph. In numerical experiments ranging from moderately dense to sparse scenarios, we obtain that, as the dimension increases, the link strength we simulate is stable with respect to the structure sparsity. Importantly, we show in a real validation setting how structure recovery is greatly improved for all learning algorithms when using our proposed method, thereby producing a more realistic comparison framework.
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