The graphical lasso: New insights and alternatives

By Rahul Mazumder, Trevor J. Hastie. Electronic Journal of Statistics, 2011

306 citations recorded by the source corpus.

The graphical lasso [5] is an algorithm for learning the structure in an undirected Gaussian graphical model, using ℓ1 regularization to control the number of zeros in the precision matrix Θ = Σ−1 [2, 11]. The R package GLASSO [5] is popular, fast, and allows one to efficiently build a path of models for different values of the tuning parameter. Convergence of GLASSO can be tricky; the converged precision matrix might not be the inverse of the estimated covariance, and occasionally it fails to converge with warm starts. In this paper we explain this behavior, and propose new algorithms that appear to outperform GLASSO. By studying the “normal equations” we see that, GLASSO is solving the dual of the graphical lasso penalized likelihood, by block coordinate ascent; a result which can also be found in [2]. In this dual, the target of estimation is Σ, the covariance matrix, rather than the precision matrix Θ. We propose similar primal algorithms P-GLASSO and DP-GLASSO, that also operate by block-coordinate descent, where Θ is the optimization target. We study all of these algorithms, and in particular different approaches to solving their coordinate sub-problems. We conclude that DP-GLASSO is superior from several points of view.

Subfields: Artificial Intelligence, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/cce262acee7b6dd9d299d0cb5e441e8b62101e8d

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Citations in the corpus

Structure Learning in Graphical Modeling (2016, methodology): “The coordinate-descent algorithm of Friedman et al. (2008) is a popular method for computation of ˆ K gl ; see Mazumder & Hastie (2012b) for a discussion of its properties.”
Benchmarking functional connectome‐based predictive models for resting‐state fMRI (2019, unknown): “The convergence problem iswell understood theoretically: the algorithm is a primal-dual algorithm and errors can accumulate between the primal and the dual solution (Mazumder and Hastie, 2012).”
The huge Package for High-dimensional Undirected Graph Estimation in R (2012, unknown): “The package supports two modes of screening, lossless [Witten et al., 2011, Mazumder and Hastie, 2011a] and lossy screening.”
High‐dimensional covariance matrix estimation (2019, background): “…cholesky decomposition (see 2.6) with regularization (Pan and Mackenzie, 2003, Pourahmadi, 2007, Rothman et al., 2010), graphical lasso (Friedman et al., 2008, Mazumder and Hastie, 2012), low rank plus sparse decomposition (Fan et al., 2008, 2013, Guo et al., 2017), to name but a few.”
A symmetric multivariate leakage correction for MEG connectomes (2015, unknown): “MEG connectivity studies which seek to be robust to source leakage artefacts either assess metrics of phase lag between sources, which are insensitive to linear leakage effects Nolte et al. (2004), Stam et al.”
Graph Learning From Data Under Laplacian and Structural Constraints (2016, methodology): “Previously, numerous BCD-type algorithms are proposed for sparse inverse covariance estimation [13][33][34] which iteratively solve an `1-regularized quadratic program.”
Honest confidence regions and optimality in high-dimensional precision matrix estimation (2015, methodology): “In comparison, the graphical Lasso method presents a more challenging computational problem, for more details see e.g. Mazumder and Hastie (2012).”
Biotic interactions hold the key to understanding metacommunity organisation (2020, background): “Since the Graphical Lasso is expected to be sensitive to the effect of a missing predictor (Mazumder and Hastie 2012), we tested to what extent the addition of a new (initially missing) environmental component affected the structure of the empirical partial correlation networks (here, based on…”
Dynamic functional connectivity analysis reveals transient states of dysconnectivity in schizophrenia (2014, background): “…has recently been reported that the original graphical LASSO implementation might not ensure positive semi-definiteness of the estimated covariance matrix (Mazumder and Hastie, 2012) and so we verified that all of the eigenvalues of the estimated dynamic covariance matrices estimated are positive.”
How reliable are MEG resting-state connectivity metrics? (2016, methodology): “We then took the linear correlation between the logarithmof ROI power envelopes, and computed partial correlations by inverting the covariance matrix of the power envelopes, and regularising using the DP-glasso (Mazumder and Hastie, 2012).”