论文标题
图形约束下高斯分布的潜在因子分析
Latent Factor Analysis of Gaussian Distributions under Graphical Constraints
论文作者
论文摘要
我们探讨了凸优化问题的解决方案空间的代数结构约束最小痕量因素分析（CMTFA），当人口协方差矩阵$σ_x$具有额外的潜在图形约束时，即潜在的星星拓扑。特别是，我们已经表明，CMTFA可以具有排名$ 1 $或排名$ n-1 $的解决方案，而两者之间没有任何内容。排名$ 1 $解决方案的特殊情况与只有一个潜在变量捕获可观察到的所有依赖关系的情况相对应，从而产生了星形拓扑。我们发现排名$ 1 $和排名$ n-1 $解决方案的明确条件的CMTFA解决方案$σ_x$。作为建造更通用的高斯树的基本尝试，我们发现了多个群集的必要和足够条件，每个集群都有$ 1 $ CMTFA解决方案，以满足结合在一起建造高斯树的最低可能性。为了支持我们的分析结果，我们提出了一些数值，证明了我们工作的贡献的有用性。
We explore the algebraic structure of the solution space of convex optimization problem Constrained Minimum Trace Factor Analysis (CMTFA), when the population covariance matrix $Σ_x$ has an additional latent graphical constraint, namely, a latent star topology. In particular, we have shown that CMTFA can have either a rank $ 1 $ or a rank $ n-1 $ solution and nothing in between. The special case of a rank $ 1 $ solution, corresponds to the case where just one latent variable captures all the dependencies among the observables, giving rise to a star topology. We found explicit conditions for both rank $ 1 $ and rank $n- 1$ solutions for CMTFA solution of $Σ_x$. As a basic attempt towards building a more general Gaussian tree, we have found a necessary and a sufficient condition for multiple clusters, each having rank $ 1 $ CMTFA solution, to satisfy a minimum probability to combine together to build a Gaussian tree. To support our analytical findings we have presented some numerical demonstrating the usefulness of the contributions of our work.
