[Paper Review] Learning Local Dependence In Ordered Data

Guo Yu, Jacob Bien|arXiv (Cornell University)|Apr 25, 2016

Statistical Methods and Inference36 references16 citations

TL;DR

This paper proposes a convex, penalized likelihood method for learning local dependence in ordered data by estimating the inverse Cholesky factor of the covariance matrix using a hierarchical group lasso penalty. The approach enables flexible, variable-specific neighborhood sizes, yielding a sparse, symmetric, positive definite precision matrix estimator with strong theoretical guarantees for signed support recovery and estimation consistency.

ABSTRACT

In many applications, data come with a natural ordering. This ordering can often induce local dependence among nearby variables. However, in complex data, the width of this dependence may vary, making simple assumptions such as a constant neighborhood size unrealistic. We propose a framework for learning this local dependence based on estimating the inverse of the Cholesky factor of the covariance matrix. Penalized maximum likelihood estimation of this matrix yields a simple regression interpretation for local dependence in which variables are predicted by their neighbors. Our proposed method involves solving a convex, penalized Gaussian likelihood problem with a hierarchical group lasso penalty. The problem decomposes into independent subproblems which can be solved efficiently in parallel using first-order methods. Our method yields a sparse, symmetric, positive definite estimator of the precision matrix, encoding a Gaussian graphical model. We derive theoretical results not found in existing methods attaining this structure. In particular, our conditions for signed support recovery and estimation consistency rates in multiple norms are as mild as those in a regression problem. Empirical results show our method performing favorably compared to existing methods. We apply our method to genomic data to flexibly model linkage disequilibrium. Our method is also applied to improve the performance of discriminant analysis in sound recording classification.

Motivation & Objective

• To address the challenge of modeling local dependence in high-dimensional ordered data, such as time series or genomic sequences, where dependence is expected to be localized but the neighborhood size varies per variable.
• To develop a method that flexibly estimates variable-specific neighborhood sizes rather than assuming a fixed or global dependence width.
• To ensure the resulting precision matrix estimator is sparse, symmetric, and positive definite, preserving the structure of a Gaussian graphical model.
• To derive theoretical guarantees—specifically signed support recovery and estimation consistency—under mild conditions comparable to those in standard regression problems.
• To enable efficient computation via decomposition into independent subproblems solvable in parallel using first-order methods.

Proposed method

• The method estimates the inverse of the Cholesky factor of the covariance matrix, which provides a regression-based interpretation: each variable is predicted by its neighboring variables in the ordering.
• A hierarchical group lasso penalty is applied to the Cholesky factor to induce sparsity and enforce local dependence structure, where each variable depends only on its closest predecessors.
• The optimization problem is convex and decomposes into independent subproblems per variable, enabling efficient parallel computation using first-order methods.
• The resulting estimator is guaranteed to be symmetric and positive definite, ensuring a valid precision matrix for Gaussian graphical models.
• The method leverages the Cholesky decomposition to naturally encode conditional independence via zero entries in the Cholesky factor, with sparsity patterns reflecting local dependence.
• Theoretical analysis establishes conditions for signed support recovery and estimation consistency in multiple norms, with rates as favorable as in standard regression settings.

Experimental results

Research questions

• RQ1 Can a convex optimization framework be developed to learn variable-specific local dependence structures in ordered data, without assuming a fixed neighborhood size?
• RQ2 How can sparsity in the precision matrix be induced while preserving symmetry and positive definiteness in the context of ordered variables?
• RQ3 What theoretical guarantees—such as signed support recovery and estimation consistency—can be established for such a method under minimal assumptions?
• RQ4 How does the performance of this method compare to existing approaches in terms of accuracy and computational efficiency on real-world data?
• RQ5 Can the method effectively model complex dependence structures such as linkage disequilibrium in genomics and improve downstream tasks like discriminant analysis in audio classification?

Key findings

• The proposed method achieves signed support recovery and estimation consistency in multiple norms under conditions as mild as those in standard regression problems, demonstrating strong theoretical robustness.
• The method yields a sparse, symmetric, and positive definite precision matrix estimator, which is essential for valid Gaussian graphical models and downstream inference.
• Empirical results show favorable performance compared to existing methods on both simulated and real data, particularly in capturing flexible local dependence patterns.
• The method was successfully applied to genomic data to model linkage disequilibrium, demonstrating its ability to capture biologically relevant dependence structures.
• In sound recording classification, the method improved discriminant analysis performance by better modeling local temporal dependencies in the data.
• The optimization problem decomposes into independent subproblems, enabling efficient parallel computation and scalability to high-dimensional settings.