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Register a new userAn efficient ADMM algorithm for high dimensional precision matrix estimation via penalized quadratic loss
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Added by
Cheng Wang
Publication date
2018
fields
Mathematical Statistics
and research's language is
English
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The estimation of high dimensional precision matrices has been a central topic in statistical learning. However, as the number of parameters scales quadratically with the dimension $p$, many state-of-the-art methods do not scale well to solve problems with a very large $p$. In this paper, we propose a very efficient algorithm for precision matrix estimation via penalized quadratic loss functions. Under the high dimension low sample size setting, the computation complexity of our algorithm is linear in both the sample size and the number of parameters. Such a computation complexity is in some sense optimal, as it is the same as the complexity needed for computing the sample covariance matrix. Numerical studies show that our algorithm is much more efficient than other state-of-the-art methods when the dimension $p$ is very large.
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In this paper, we estimate the high dimensional precision matrix under the weak sparsity condition where many entries are nearly zero. We study a Lasso-type method for high dimensional precision matrix estimation and derive general error bounds under the weak sparsity condition. The common irrepresentable condition is relaxed and the results are applicable to the weak sparse matrix. As applications, we study the precision matrix estimation for the heavy-tailed data, the non-paranormal data, and the matrix data with the Lasso-type method.
Maximum likelihood estimation of mixture proportions has a long history, and continues to play an important role in modern statistics, including in development of nonparametric empirical Bayes methods. Maximum likelihood of mixture proportions has traditionally been solved using the expectation maximization (EM) algorithm, but recent work by Koenker & Mizera shows that modern convex optimization techniques -- in particular, interior point methods -- are substantially faster and more accurate than EM. Here, we develop a new solution based on sequential quadratic programming (SQP). It is substantially faster than the interior point method, and just as accurate. Our approach combines several ideas: first, it solves a reformulation of the original problem; second, it uses an SQP approach to make the best use of the expensive gradient and Hessian computations; third, the SQP iterations are implemented using an active set method to exploit the sparse nature of the quadratic subproblems; fourth, it uses accurate low-rank approximations for more efficient gradient and Hessian computations. We illustrate the benefits of our approach in experiments on synthetic data sets as well as a large genetic association data set. In large data sets (n = 1,000,000 observations, m = 1,000 mixture components), our implementation achieves at least 100-fold reduction in runtime compared with a state-of-the-art interior point solver. Our methods are implemented in Julia, and in an R package available on CRAN (see https://CRAN.R-project.org/package=mixsqp).
Estimation of the precision matrix (or inverse covariance matrix) is of great importance in statistical data analysis. However, as the number of parameters scales quadratically with the dimension p, computation becomes very challenging when p is large. In this paper, we propose an adaptive sieving reduction algorithm to generate a solution path for the estimation of precision matrices under the $ell_1$ penalized D-trace loss, with each subproblem being solved by a second-order algorithm. In each iteration of our algorithm, we are able to greatly reduce the number of variables in the problem based on the Karush-Kuhn-Tucker (KKT) conditions and the sparse structure of the estimated precision matrix in the previous iteration. As a result, our algorithm is capable of handling datasets with very high dimensions that may go beyond the capacity of the existing methods. Moreover, for the sub-problem in each iteration, other than solving the primal problem directly, we develop a semismooth Newton augmented Lagrangian algorithm with global linear convergence on the dual problem to improve the efficiency. Theoretical properties of our proposed algorithm have been established. In particular, we show that the convergence rate of our algorithm is asymptotically superlinear. The high efficiency and promising performance of our algorithm are illustrated via extensive simulation studies and real data applications, with comparison to several state-of-the-art solvers.
The smoothly clipped absolute deviation (SCAD) and the minimax concave penalty (MCP) penalized regression models are two important and widely used nonconvex sparse learning tools that can handle variable selection and parameter estimation simultaneously, and thus have potential applications in various fields such as mining biological data in high-throughput biomedical studies. Theoretically, these two models enjoy the oracle property even in the high-dimensional settings, where the number of predictors $p$ may be much larger than the number of observations $n$. However, numerically, it is quite challenging to develop fast and stable algorithms due to their non-convexity and non-smoothness. In this paper we develop a fast algorithm for SCAD and MCP penalized learning problems. First, we show that the global minimizers of both models are roots of the nonsmooth equations. Then, a semi-smooth Newton (SSN) algorithm is employed to solve the equations. We prove that the SSN algorithm converges locally and superlinearly to the Karush-Kuhn-Tucker (KKT) points. Computational complexity analysis shows that the cost of the SSN algorithm per iteration is $O(np)$. Combined with the warm-start technique, the SSN algorithm can be very efficient and accurate. Simulation studies and a real data example suggest that our SSN algorithm, with comparable solution accuracy with the coordinate descent (CD) and the difference of convex (DC) proximal Newton algorithms, is more computationally efficient.
Quadratic regression goes beyond the linear model by simultaneously including main effects and interactions between the covariates. The problem of interaction estimation in high dimensional quadratic regression has received extensive attention in the past decade. In this article we introduce a novel method which allows us to estimate the main effects and interactions separately. Unlike existing methods for ultrahigh dimensional quadratic regressions, our proposal does not require the widely used heredity assumption. In addition, our proposed estimates have explicit formulas and obey the invariance principle at the population level. We estimate the interactions of matrix form under penalized convex loss function. The resulting estimates are shown to be consistent even when the covariate dimension is an exponential order of the sample size. We develop an efficient ADMM algorithm to implement the penalized estimation. This ADMM algorithm fully explores the cheap computational cost of matrix multiplication and is much more efficient than existing penalized methods such as all pairs LASSO. We demonstrate the promising performance of our proposal through extensive numerical studies.
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