6,954 research outputs found
Sparse Matrix Inversion with Scaled Lasso
We propose a new method of learning a sparse nonnegative-definite target
matrix. Our primary example of the target matrix is the inverse of a population
covariance or correlation matrix. The algorithm first estimates each column of
the target matrix by the scaled Lasso and then adjusts the matrix estimator to
be symmetric. The penalty level of the scaled Lasso for each column is
completely determined by data via convex minimization, without using
cross-validation.
We prove that this scaled Lasso method guarantees the fastest proven rate of
convergence in the spectrum norm under conditions of weaker form than those in
the existing analyses of other regularized algorithms, and has faster
guaranteed rate of convergence when the ratio of the and spectrum
norms of the target inverse matrix diverges to infinity. A simulation study
demonstrates the computational feasibility and superb performance of the
proposed method.
Our analysis also provides new performance bounds for the Lasso and scaled
Lasso to guarantee higher concentration of the error at a smaller threshold
level than previous analyses, and to allow the use of the union bound in
column-by-column applications of the scaled Lasso without an adjustment of the
penalty level. In addition, the least squares estimation after the scaled Lasso
selection is considered and proven to guarantee performance bounds similar to
that of the scaled Lasso
Group Symmetry and non-Gaussian Covariance Estimation
We consider robust covariance estimation with group symmetry constraints.
Non-Gaussian covariance estimation, e.g., Tyler scatter estimator and
Multivariate Generalized Gaussian distribution methods, usually involve
non-convex minimization problems. Recently, it was shown that the underlying
principle behind their success is an extended form of convexity over the
geodesics in the manifold of positive definite matrices. A modern approach to
improve estimation accuracy is to exploit prior knowledge via additional
constraints, e.g., restricting the attention to specific classes of covariances
which adhere to prior symmetry structures. In this paper, we prove that such
group symmetry constraints are also geodesically convex and can therefore be
incorporated into various non-Gaussian covariance estimators. Practical
examples of such sets include: circulant, persymmetric and complex/quaternion
proper structures. We provide a simple numerical technique for finding maximum
likelihood estimates under such constraints, and demonstrate their performance
advantage using synthetic experiments
Regularized Covariance Matrix Estimation in Complex Elliptically Symmetric Distributions Using the Expected Likelihood Approach - Part 2: The Under-Sampled Case
In the first part of this series of two papers, we extended the expected likelihood approach originally developed in the Gaussian case, to the broader class of complex elliptically symmetric (CES) distributions and complex angular central Gaussian (ACG) distributions. More precisely, we demonstrated that the probability density function (p.d.f.) of the likelihood ratio (LR) for the (unknown) actual scatter matrix \mSigma_{0} does not depend on the latter: it only depends on the density generator for the CES distribution and is distribution-free in the case of ACG distributed data, i.e., it only depends on the matrix dimension and the number of independent training samples , assuming that . Additionally, regularized scatter matrix estimates based on the EL methodology were derived. In this second part, we consider the under-sampled scenario () which deserves a specific treatment since conventional maximum likelihood estimates do not exist. Indeed, inference about the scatter matrix can only be made in the -dimensional subspace spanned by the columns of the data matrix. We extend the results derived under the Gaussian assumption to the CES and ACG class of distributions. Invariance properties of the under-sampled likelihood ratio evaluated at \mSigma_{0} are presented. Remarkably enough, in the ACG case, the p.d.f. of this LR can be written in a rather simple form as a product of beta distributed random variables. The regularized schemes derived in the first part, based on the EL principle, are extended to the under-sampled scenario and assessed through numerical simulations
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