A Constrained ℓ1 Minimization Approach to Sparse Precision Matrix Estimation

By Tommaso Cai, Weidong Liu, Xi Luo. 2011

1,164 citations recorded by the source corpus.

This article proposes a constrained ℓ1 minimization method for estimating a sparse inverse covariance matrix based on a sample of n iid p-variate random variables. The resulting estimator is shown to have a number of desirable properties. In particular, the rate of convergence between the estimator and the true s-sparse precision matrix under the spectral norm is when the population distribution has either exponential-type tails or polynomial-type tails. We present convergence rates under the elementwise ℓ∞ norm and Frobenius norm. In addition, we consider graphical model selection. The procedure is easily implemented by linear programming. Numerical performance of the estimator is investigated using both simulated and real data. In particular, the procedure is applied to analyze a breast cancer dataset and is found to perform favorably compared with existing methods.

Subfields: Computational Mechanics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/d06798476ae907160dc4ec6b88be466fdcdb8b6d

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Computational Mechanics

In Statistics and Probability

Citations in the corpus

Estimating structured high-dimensional covariance and precision matrices: Optimal rates and adaptive estimation (2016, background): “Over the class HP(cn,p,M), the estimators proposed in Yuan (2010), Cai et al. (2011) and Cai et al. (2012) can be shown to attain the optimal rate under the spectral norm if the matrix 1 norm Mn,p is bounded.”
Estimating Sparse Precision Matrix: Optimal Rates of Convergence and Adaptive Estimation (2012, background): “For 1 ≤ i ≤ p, let ω̂i be the solution of the following convex optimization problem min |ω|1 subject to |Σnω − ei|∞ ≤ τn, (5) 5”
A Projection-based Conditional Dependence Measure with Applications to High-dimensional Undirected Graphical Models. (2015, background): “…and high-dimensional settings, including Dempster (1972), Drton and Perlman (2004), Meinshausen and Bühlmann (2006), Friedman et al. (2008), Fan et al. (2009), Cai et al. (2011), Liu (2013), Chen et al. (2014), Ren et al. (2015), Jankova and Van De Geer (2015) and Yu and Bien (2017).”
The huge Package for High-dimensional Undirected Graph Estimation in R (2012, positive): “[2011], Mazumder and Hastie [2011b] and the lossy screening method is from Fan and Lv [2008]. Such screening procedures can greatly reduce the computational cost and achieve equal or even better estimation by reducing the variance at the expense of an increase in bias.”
Likelihood Ratio Tests for Dependent Data with Applications to Longitudinal and Functional Data Analysis (2014, background): “This condition (see also Cai et al., 2011) is related to the rate of convergence between the inverse covariance estimator and the true precision matrix in the spectral norm.”
Structure Learning in Graphical Modeling (2016, background): “For example, Cai et al. (2011) minimize (cid:107) K (cid:107) 1 subject to a constraint on (cid:107) SK − I (cid:107) ∞ .”
STRONG ORACLE OPTIMALITY OF FOLDED CONCAVE PENALIZED ESTIMATION (2012, methodology): “Similar results hold under the exponential/polynomial tail condition in Cai et al. (2011).”
Performance bounds for parameter estimates of high-dimensional linear models with correlated errors (2016, methodology): “Following [8], we call ˆ β the Clime estimator .”
Estimation and Inference about Heterogeneous Treatment Effects in High-Dimensional Dynamic Panels. (2017, methodology): “It extends previously available results of Cai et al. (2011) to allow for the (1) biased first-stage estimation error in the empirical covariance matrix Q̂ = ENT V̂itV̂ ′ it and (2) weak dependence across time periods.”