High-dimensional graphs and variable selection with the Lasso
Open accessAbstract
The pattern of zero entries in the inverse covariance matrix of a multivariate normal distribution corresponds to conditional independence restrictions between variables. Covariance selection aims at estimating those structural zeros from data. We show that neighborhood selection with the Lasso is a computationally attractive alternative to standard covariance selection for sparse high-dimensional graphs. Neighborhood selection estimates the conditional independence restrictions separately for each node in the graph and is hence equivalent to variable selection for Gaussian linear models. We show that the proposed neighborhood selection scheme is consistent for sparse high-dimensional graphs. Consistency hinges on the choice of the penalty parameter. The oracle value for optimal prediction does not lead to a consistent neighborhood estimate. Controlling instead the probability of falsely joining some distinct connectivity components of the graph, consistent estimation for sparse graphs is achieved (with exponential rates), even when the number of variables grows as the number of observations raised to an arbitrary power.
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Lasso-regularized covariance selection methods can recover the off-diagonal structure of the covariance matrix.
“These methods have the advantage to be able to recover the off diagonal structure of the covariance matrix.”
From Signal Estimation and Uncertainties Extraction in Terahertz Time-Domain Spectroscopy (2025) · cites this paper as [57]
The Abstract identifies the structural zeros targeted by covariance selection as entries of the inverse covariance matrix, not the covariance matrix. Section 3 says Lasso neighborhood selection recovers the graph’s edge set, which represents conditional-independence structure; it does not claim recovery of the covariance matrix’s off-diagonal structure. Page numbers are unavailable in our copy. (LLM, conf=0.87)
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“There are also methods that rely on solving the covariance selection problem to approximate sparse inverse covariance matrices with lasso regularization [57], [58], [59].”
- Authors
- Meinshausen, N., & Bühlmann, P.
- Affiliations
- ETH Zurich (CH)
- ORCID
- Bühlmann, Peter
- Venue
- The Annals of Statistics
- Publisher
- Institute of Mathematical Statistics
- Published
- 2006-06-01
- Volume
- 34
- Issue
- 3
- Pages
- 1436-1462
- ISSN
- 0090-5364 (ISSN-L)
- 2168-8966
- OpenAlex
- W2010824638
- Semantic Scholar
- 144feb8f6b2bd36775ef6cc39452387c671314ff
- Cited by
- 2,521
- References
- 13
- Citations by year
- 2026: 91 · 2025: 162 · 2024: 153 · 2023: 123 · 2022: 207 · 2021: 166 · 2020: 182 · 2019: 199 · 2018: 171 · 2017: 155 · 2016: 154 · 2015: 117 · 2014: 133 · 2013: 103 · 2012: 114
- FWCI
- 41.93
- Type
- journal article
- Language
- en
- Open access
- bronze
- doi.org
- Locations
- The Annals of Statistics (publishedVersion)
- arXiv (Cornell University) (submittedVersion)
- Project Euclid (Cornell University) (submittedVersion)
- arXiv (Cornell University)
- Topics
- Physical Sciences › Mathematics › Statistics and Probability › Statistical Methods and Inference
- Physical Sciences › Engineering › Computational Mechanics › Sparse and Compressive Sensing Techniques
- Physical Sciences › Computer Science › Artificial Intelligence › Bayesian Modeling and Causal Inference
- Keywords
- neighborhood selection, LASSO, covariance selection, high-dimensional graphs, sparse graphs, feature selection, conditional independence, inverse covariance matrix, consistency, penalty parameter selection, oracle inequalities, exponential rate, linear Gaussian models
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