大型自反广义逆矩阵与Moore-Penrose逆矩阵的谱分析

Some comments about measures, Jacobians and Moore-Penrose inverse

Some general problems of Jacobian computations in non-full rank matrices are discussed in this work. In particular, the Jacobian of the Moore-Penrose inverse derived via matrix differential calculus is revisited. Then the Jacobian in the full rank case is derived under the simple and old theory of the exterior product.

Spectral analysis of the Moore-Penrose inverse of a large dimensional sample covariance matrix

For a sample of $n$ independent identically distributed $p$-dimensional centered random vectors with covariance matrix $\mathbfΣ_n$ let $\tilde{\mathbf{S}}_n$ denote the usual sample covariance (centered by the mean) and $\mathbf{S}_n$ the non-centered sample covariance matrix (i.e. the matrix of second moment estimates), where $p> n$. In this paper, we provide the limiting spectral distribution and central limit theorem for linear spectral statistics of the Moore-Penrose inverse of $\mathbf{S}_n$ and $\tilde{\mathbf{S}}_n$. We consider the large dimensional asymptotics when the number of vari…

An Interpretation of the Moore-Penrose Generalized Inverse of a Singular Fisher Information Matrix

It is proved that in a non-Bayesian parametric estimation problem, if the Fisher information matrix (FIM) is singular, unbiased estimators for the unknown parameter will not exist. Cramer-Rao bound (CRB), a popular tool to lower bound the variances of unbiased estimators, seems inapplicable in such situations. In this paper, we show that the Moore-Penrose generalized inverse of a singular FIM can be interpreted as the CRB corresponding to the minimum variance among all choices of minimum constraint functions. This result ensures the logical validity of applying the Moore-Penrose generalized in…

复兴伪逆：大规模维度下Moore-Penrose和岭型逆的渐近性质及其应用

Reviving pseudo-inverses: Asymptotic properties of large dimensional Moore-Penrose and Ridge-type inverses with applications

本文研究了大规模维度下Moore-Penrose和岭型逆的渐近性质，提出了一种通用方法用于构建改进的收缩估计器，适用于精度矩阵和投资组合权重等应用。

1-norm minimization and minimum-rank structured sparsity for symmetric and ah-symmetric generalized inverses: rank one and two

Generalized inverses are important in statistics and other areas of applied matrix algebra. A \emph{generalized inverse} of a real matrix $A$ is a matrix $H$ that satisfies the Moore-Penrose (M-P) property $AHA=A$. If $H$ also satisfies the M-P property $HAH=H$, then it is called \emph{reflexive}. Reflexivity of a generalized inverse is equivalent to minimum rank, a highly desirable property. We consider aspects of symmetry related to the calculation of various \emph{sparse} reflexive generalized inverses of $A$. As is common, we use (vector) 1-norm minimization for both inducing sparsity and …