G-AMA: Sparse Gaussian graphical model estimation via alternating minimization

Bala Rajaratnam; Onkar Dalal

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G-AMA: Sparse Gaussian graphical model estimation via alternating minimization

Onkar Dalal, Bala Rajaratnam

Computation · Submitted 13 May 2014

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim The methodology is demonstrated on both simulated and real datasets to illustrate its superior performance over recently proposed methods.

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What the abstract itself states, as probabilities: claims first 0.06 · states limitation 0.06 · human participants 0.10 · controlled comparison 0.29 · randomized assignment 0.01 · states quantitative result 0.12 · compares with prior results 0.97 · mentions code or data availability 0.02.

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Several methods have been recently proposed for estimating sparse Gaussian graphical models using $\ell_{1}$ regularization on the inverse covariance matrix. Despite recent advances, contemporary applications require methods that are even faster in order to handle ill-conditioned high dimensional modern day datasets. In this paper, we propose a new method, G-AMA, to solve the sparse inverse covariance estimation problem using Alternating Minimization Algorithm (AMA), that effectively works as a proximal gradient algorithm on the dual problem. Our approach has several novel advantages over existing methods. First, we demonstrate that G-AMA is faster than the previous best algorithms by many orders of magnitude and is thus an ideal approach for modern high throughput applications. Second, global linear convergence of G-AMA is demonstrated rigorously, underscoring its good theoretical properties. Third, the dual algorithm operates on the covariance matrix, and thus easily facilitates incorporating additional constraints on pairwise/marginal relationships between feature pairs based on domain specific knowledge. Over and above estimating a sparse inverse covariance matrix, we also illustrate how to (1) incorporate constraints on the (bivariate) correlations and, (2) incorporate equality (equisparsity) or linear constraints between individual inverse covariance elements. Fourth, we also show that G-AMA is better adept at handling extremely ill-conditioned problems, as is often the case with real data. The methodology is demonstrated on both simulated and real datasets to illustrate its superior performance over recently proposed methods.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	Several methods have been recently proposed for estimating sparse Gaussian graphical models using $\ell_{1}$ regularization on the inverse covariance matrix.	background 1.00	0.04	0.02	asserted 1.00
s1	Despite recent advances, contemporary applications require methods that are even faster in order to handle ill-conditioned high dimensional modern day datasets.	gap 0.93	0.06	0.03	asserted 0.99
s2	In this paper, we propose a new method, G-AMA, to solve the sparse inverse covariance estimation problem using Alternating Minimization Algorithm (AMA), that effectively works as a proximal gradient algorithm on the dual problem.	contribution 0.75	0.89	0.19	asserted 1.00
s3	Our approach has several novel advantages over existing methods.	contribution 0.98	0.37	0.07	asserted 0.98
s4	First, we demonstrate that G-AMA is faster than the previous best algorithms by many orders of magnitude and is thus an ideal approach for modern high throughput applications.	result 0.87	0.90	0.39	asserted 1.00
s5	Second, global linear convergence of G-AMA is demonstrated rigorously, underscoring its good theoretical properties.	result 0.58	0.95	0.11	asserted 1.00
s6	Third, the dual algorithm operates on the covariance matrix, and thus easily facilitates incorporating additional constraints on pairwise/marginal relationships between feature pairs based on domain specific knowledge.	contribution 0.94	0.76	0.06	asserted 0.98
s7	Over and above estimating a sparse inverse covariance matrix, we also illustrate how to (1) incorporate constraints on the (bivariate) correlations and, (2) incorporate equality (equisparsity) or linear constraints between individual inverse covariance elements.	contribution 0.95	0.91	0.07	asserted 1.00
s8	Fourth, we also show that G-AMA is better adept at handling extremely ill-conditioned problems, as is often the case with real data.	result 0.54	0.89	0.10	asserted 1.00
s9	The methodology is demonstrated on both simulated and real datasets to illustrate its superior performance over recently proposed methods.	result 0.69	0.93	0.26	asserted 1.00

What kind of paper the classifier reports this is: method or tool 1.00 · evidence from simulation or computation 0.90.

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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

The paper

Original abstract

Several methods have been recently proposed for estimating sparse Gaussian graphical models using $\ell_{1}$ regularization on the inverse covariance matrix. Despite recent advances, contemporary applications require methods that are even faster in order to handle ill-conditioned high dimensional modern day datasets. In this paper, we propose a new method, G-AMA, to solve the sparse inverse covariance estimation problem using Alternating Minimization Algorithm (AMA), that effectively works as a proximal gradient algorithm on the dual problem. Our approach has several novel advantages over existing methods. First, we demonstrate that G-AMA is faster than the previous best algorithms by many orders of magnitude and is thus an ideal approach for modern high throughput applications. Second, global linear convergence of G-AMA is demonstrated rigorously, underscoring its good theoretical properties. Third, the dual algorithm operates on the covariance matrix, and thus easily facilitates incorporating additional constraints on pairwise/marginal relationships between feature pairs based on domain specific knowledge. Over and above estimating a sparse inverse covariance matrix, we also illustrate how to (1) incorporate constraints on the (bivariate) correlations and, (2) incorporate equality (equisparsity) or linear constraints between individual inverse covariance elements. Fourth, we also show that G-AMA is better adept at handling extremely ill-conditioned problems, as is often the case with real data. The methodology is demonstrated on both simulated and real datasets to illustrate its superior performance over recently proposed methods.

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3 arXiv papers cite this in 5 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

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Uses its method or tool 1 paper

We follow an enhanced variant of AMA [ref] which utilizes an adaptive BB step-size selection [ref] in [ref], [ref], and [ref] to guarantee sufficient dual ascent and positive definiteness of $X$.

Low-complexity modeling of partially available second-order statistics: theory and an efficient matrix completion algorithm · Choice of step-size for the dual update [ref], [ref] · 0.97

Cites it as background 2 papers

We note that AMA was originally developed by Tseng [ref] and its enhanced variants have been recently presented in [ref] and used, in particular, for estimation of sparse Gaussian graphical models.

Low-complexity modeling of partially available second-order statistics: theory and an efficient matrix completion algorithm · Customized algorithm for solving the covariance completion problem · 0.81
The literature on this topic is simply enormous and for the sake of brevity, space constraints and the topic of this paper, we avoid an extensive literature review by referring to the references in the seminal work of [ref] and the very recent work of [ref] .

Optimization Methods for Sparse Pseudo-Likelihood Graphical Model Selection · Background · 1.00

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Builds on 1 paper

This algorithm was later generalized in [ref] .

Proximal Newton-type methods for minimizing composite functions · Proximal Gradient and Proximal Newton-like Methods · 0.80

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We also note that the G-AMA algorithm is different from the ALM method [ref] in several fundamental aspects, including (a) Theoretical, (b) Algorithmic and, (c) Numerical.

Sparse Inverse Covariance Selection via Alternating Linearization Methods · Connections to ALM CITEMARK25Z · 0.99

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