Abstract Factor models have been proposed for a broad range of observed variables such as binary, Gaussian, and variables of mixed types. They typically model a pairwise interaction parameter matrix with a low-rank and a diagonal component. The low-rank component can be interpreted as the effect of a few latent quantitative variables which are common in social science applications. Another line of research has investigated graphical models for the same types of observed variables, where the pairwise interactions are usually assumed to be sparse. Sparse network structures are often found in the natural sciences. Still overall, while factor and sparse models are suitable for many applications, they sometimes might not be expressive enough to fit certain data well. This has motivated the confluence of factor and graphical models, yielding models where the interaction parameters are decomposed into respectively sparse and low-rank components. Up to now, this has only been done separately for observed Gaussian and for observed binary variables, but never jointly. The present work accomplishes a unified treatment of pairwise sparse and low-rank models. It does so by simultaneously allowing observed binary and quantitative (conditional Gaussian) variables. The model parameters of the joint model can be estimated using a convex regularized likelihood optimization problem. We show that the resulting estimator has consistency properties in the high-dimensional setting.

中文翻译：

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混合类型变量的成对稀疏 + 低秩模型

摘要 因子模型已被提出用于广泛的观察变量，例如二进制、高斯和混合类型的变量。他们通常对具有低秩和对角线分量的成对交互参数矩阵进行建模。低秩成分可以解释为社会科学应用中常见的一些潜在定量变量的影响。另一项研究调查了相同类型的观察变量的图形模型，其中通常假设成对相互作用是稀疏的。稀疏网络结构经常出现在自然科学中。总体而言，虽然因子模型和稀疏模型适用于许多应用程序，但它们有时可能不够表达，无法很好地拟合某些数据。这激发了因子和图形模型的融合，生成模型，其中交互参数分别分解为稀疏和低秩分量。到目前为止，这仅针对观察到的高斯变量和观察到的二元变量单独完成，但从未联合完成。目前的工作完成了对成对稀疏和低秩模型的统一处理。它通过同时允许观察到的二进制和定量（条件高斯）变量来实现。联合模型的模型参数可以使用凸正则化似然优化问题来估计。我们表明，由此产生的估计器在高维设置中具有一致性属性。目前的工作完成了对成对稀疏和低秩模型的统一处理。它通过同时允许观察到的二进制和定量（条件高斯）变量来实现。联合模型的模型参数可以使用凸正则化似然优化问题来估计。我们表明，由此产生的估计器在高维设置中具有一致性属性。目前的工作完成了对成对稀疏和低秩模型的统一处理。它通过同时允许观察到的二进制和定量（条件高斯）变量来实现。联合模型的模型参数可以使用凸正则化似然优化问题来估计。我们表明，由此产生的估计器在高维设置中具有一致性属性。