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.
Khoury College of Computer Sciences（科瑞计算机学院）；Northeastern University（东北大学）；National University of Singapore（新加坡国立大学）
Graphical model selection in Markov random fields is a fundamental problem in statistics and machine learning. Two particularly prominent models, the Ising model and Gaussian model, have largely developed in parallel using different (though often related) techniques, and several practical algorithms with rigorous sample complexity bounds have been established for each. In this paper, we adapt a recently proposed algorithm of Klivans and Meka (FOCS, 2017), based on the method of multiplicative weight updates, from the Ising model to the Gaussian model, via non-trivial modifications to both the …
Université Paris-Est（巴黎东大学）；Ecole des Ponts（国立桥路学校）
The edge structure of the graph defining an undirected graphical model describes precisely the structure of dependence between the variables in the graph. In many applications, the dependence structure is unknown and it is desirable to learn it from data, often because it is a preliminary step to be able to ascertain causal effects. This problem, known as structure learning, is hard in general, but for Gaussian graphical models it is slightly easier because the structure of the graph is given by the sparsity pattern of the precision matrix of the joint distribution, and because independence co…
We focus on the problem of estimating the change in the dependency structures of two $p$-dimensional Gaussian Graphical models (GGMs). Previous studies for sparse change estimation in GGMs involve expensive and difficult non-smooth optimization. We propose a novel method, DIFFEE for estimating DIFFerential networks via an Elementary Estimator under a high-dimensional situation. DIFFEE is solved through a faster and closed form solution that enables it to work in large-scale settings. We conduct a rigorous statistical analysis showing that surprisingly DIFFEE achieves the same asymptotic conver…
In this paper, we explore the algebraic structures of solution spaces for Gaussian latent factor analysis when the population covariance matrix $Σ_x$ has an additional latent graphical constraint, namely, a latent star topology. In particular, we give sufficient and necessary conditions under which the solutions to constrained minimum trace factor analysis (CMTFA) is still star. We further show that the solution to CMTFA under the star constraint can only have two cases, i.e. the number of latent variable can be only one (star) or $n-1$ where $n$ is the dimension of the observable vector, and …
University of Hawaii（夏威夷大学）；John A. Burns School of Medicine（约翰·A·伯恩斯医学院）
A classic application of description length is for model selection with the minimum description length (MDL) principle. The focus of this paper is to extend description length for data analysis beyond simple model selection and sequences of scalars. More specifically, we extend the description length for data analysis in Gaussian graphical models. These are powerful tools to model interactions among variables in a sequence of i.i.d Gaussian data in the form of a graph. Our method uses universal graph coding methods to accurately account for model complexity, and therefore provide a more rigoro…
The Constrained Minimum Determinant Factor Analysis (CMDFA) setting was motivated by Wyner's common information problem where we seek a latent representation of a given Gaussian vector distribution with the minimum mutual information under certain generative constraints. In this paper, we explore the algebraic structures of the solution space of the CMDFA, when the underlying covariance matrix $Σ_x$ has an additional latent graphical constraint, namely, a latent star topology. In particular, sufficient and necessary conditions in terms of the relationships between edge weights of the star grap…
The Constrained Minimum Determinant Factor Analysis (CMDFA) setting was motivated by Wyner's common information problem where we seek a latent representation of a given Gaussian vector distribution with the minimum mutual information under certain generative constraints. In this paper, we explore the algebraic structures of the solution space of the CMDFA, when the underlying covariance matrix $Σ_x$ has an additional latent graphical constraint, namely, a latent star topology. In particular, sufficient and necessary conditions in terms of the relationships between edge weights of the star grap…
