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Register a new userSpectral analysis of large reflexive generalized inverse and Moore-Penrose inverse matrices
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A reflexive generalized inverse and the Moore-Penrose inverse are often confused in statistical literature but in fact they have completely different behaviour in case the population covariance matrix is not a multiple of identity. In this paper, we study the spectral properties of a reflexive generalized inverse and of the Moore-Penrose inverse of the sample covariance matrix. The obtained results are used to assess the difference in the asymptotic behaviour of their eigenvalues.
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In this paper, we investigate the perturbation for the Moore-Penrose inverse of closed operators on Hilbert spaces. By virtue of a new inner product defined on $H$, we give the expression of the Moore-Penrose inverse $bar{T}^dag$ and the upper bounds of $|bar{T}^dag|$ and $|bar{T}^dag -T^dag|$. These results obtained in this paper extend and improve many related results in this area.
A matrix $P$ is said to be a nontrivial generalized reflection matrix over the real quaternion algebra $mathbb{H}$ if $P^{ast }=P eq I$ and $P^{2}=I$ where $ast$ means conjugate and transpose. We say that $Ainmathbb{H}^{ntimes n}$ is generalized reflexive (or generalized antireflexive) with respect to the matrix pair $(P,Q)$ if $A=PAQ$ $($or $A=-PAQ)$ where $P$ and $Q$ are two nontrivial generalized reflection matrices of demension $n$. Let ${large varphi}$ be one of the following subsets of $mathbb{H}^{ntimes n}$ : (i) generalized reflexive matrix; (ii)reflexive matrix; (iii) generalized antireflexive matrix; (iiii) antireflexive matrix. Let $Zinmathbb{H}^{ntimes m}$ with rank$left( Zright) =m$ and $Lambda=$ diag$left( lambda_{1},...,lambda_{m}right) .$ The inverse eigenproblem is to find a matrix $A$ such that the set ${large varphi }left( Z,Lambdaright) =left{ Ain{large varphi}text{ }|text{ }AZ=ZLambdaright} $ nonempty and find the general expression of $A.$ ewline In this paper, we investigate the inverse eigenproblem ${large varphi}left( Z,Lambdaright) $. Moreover, the approximation problem: $underset{Ain{large varphi}}{minleftVert A-ErightVert _{F}}$ is studied, where $E$ is a given matrix over $mathbb{H}$ and $parallel cdotparallel_{F}$ is the Frobenius norm.
Recovering low-rank structures via eigenvector perturbation analysis is a common problem in statistical machine learning, such as in factor analysis, community detection, ranking, matrix completion, among others. While a large variety of bounds are available for average errors between empirical and population statistics of eigenvectors, few results are tight for entrywise analyses, which are critical for a number of problems such as community detection. This paper investigates entrywise behaviors of eigenvectors for a large class of random matrices whose expectations are low-rank, which helps settle the conjecture in Abbe et al. (2014b) that the spectral algorithm achieves exact recovery in the stochastic block model without any trimming or cleaning steps. The key is a first-order approximation of eigenvectors under the $ell_infty$ norm: $$u_k approx frac{A u_k^*}{lambda_k^*},$$ where ${u_k}$ and ${u_k^*}$ are eigenvectors of a random matrix $A$ and its expectation $mathbb{E} A$, respectively. The fact that the approximation is both tight and linear in $A$ facilitates sharp comparisons between $u_k$ and $u_k^*$. In particular, it allows for comparing the signs of $u_k$ and $u_k^*$ even if $| u_k - u_k^*|_{infty}$ is large. The results are further extended to perturbations of eigenspaces, yielding new $ell_infty$-type bounds for synchronization ($mathbb{Z}_2$-spiked Wigner model) and noisy matrix completion.
Consider a standard white Wishart matrix with parameters $n$ and $p$. Motivated by applications in high-dimensional statistics and signal processing, we perform asymptotic analysis on the maxima and minima of the eigenvalues of all the $m times m$ principal minors, under the asymptotic regime that $n,p,m$ go to infinity. Asymptotic results concerning extreme eigenvalues of principal minors of real Wigner matrices are also obtained. In addition, we discuss an application of the theoretical results to the construction of compressed sensing matrices, which provides insights to compressed sensing in signal processing and high dimensional linear regression in statistics.
In this article properties of the $(b, c)$-inverse, the inverse along an element, the outer inverse with prescribed range and null space $A^{(2)}_{T, S}$ and the Moore-Penrose inverse will be studied in the contexts of Banach spaces operators, Banach algebras and $C^*$-algebras. The main properties to be considered are the continuity, the differentiability and the openness of the sets of all invertible elements defined by all the aforementioned outer inverses but the Moore-Penrose inverse. The relationship between the $(b, c)$-inverse and the outer inverse $A^{(2)}_{T, S}$ will be also characterized.
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