Topic
P-matrix
About: P-matrix is a research topic. Over the lifetime, 181 publications have been published within this topic receiving 5503 citations.
Papers published on a yearly basis
Papers
TL;DR: In this article, the authors derived the Tracey-Widom law of order 1 for large p and n matrices, where p is the largest eigenvalue of a p-variate Wishart distribution on n degrees of freedom with identity covariance.
Abstract: Let x(1) denote the square of the largest singular value of an n × p matrix X, all of whose entries are independent standard Gaussian variates. Equivalently, x(1) is the largest principal component variance of the covariance matrix $X'X$, or the largest eigenvalue of a pvariate Wishart distribution on n degrees of freedom with identity covariance. Consider the limit of large p and n with $n/p = \gamma \ge 1$. When centered by $\mu_p = (\sqrt{n-1} + \sqrt{p})^2$ and scaled by $\sigma_p = (\sqrt{n-1} + \sqrt{p})(1/\sqrt{n-1} + 1/\sqrt{p}^{1/3}$, the distribution of x(1) approaches the Tracey-Widom law of order 1, which is defined in terms of the Painleve II differential equation and can be numerically evaluated and tabulated in software. Simulations show the approximation to be informative for n and p as small as 5. The limit is derived via a corresponding result for complex Wishart matrices using methods from random matrix theory. The result suggests that some aspects of large p multivariate distribution theory may be easier to apply in practice than their fixed p counterparts.
2,599 citations
28 May 2002
TL;DR: A third approximation method based on a classic matrix completion technique that allows for principal warp analysis as a by-product is described and a significant improvement over the naive method is demonstrated.
Abstract: The thin plate spline (TPS) is an effective tool for modeling coordinate transformations that has been applied successfully in several computer vision applications. Unfortunately the solution requires the inversion of a p × p matrix, where p is the number of points in the data set, thus making it impractical for large scale applications. As it turns out, a surprisingly good approximate solution is often possible using only a small subset of corresponding points. We begin by discussing the obvious approach of using the subsampled set to estimate a transformation that is then applied to all the points, and we show the drawbacks of this method. We then proceed to borrow a technique from the machine learning community for function approximation using radial basis functions (RBFs) and adapt it to the task at hand. Using this method, we demonstrate a significant improvement over the naive method. One drawback of this method, however, is that is does not allow for principal warp analysis, a technique for studying shape deformations introduced by Bookstein based on the eigenvectors of the p × p bending energy matrix. To address this, we describe a third approximation method based on a classic matrix completion technique that allows for principal warp analysis as a by-product. By means of experiments on real and synthetic data, we demonstrate the pros and cons of these different approximations so as to allow the reader to make an informed decision suited to his or her application.
294 citations
TL;DR: In this paper, the authors extend these notions to a linear transformation defined on a Euclidean Jordan algebra and study some interconnections between these extended concepts and specialize them to the space S n of all n × n real symmetric matrices with the semidefinite cone S n and the space R n with the Lorentz cone.
232 citations
TL;DR: New error bounds are given for the linear complementarity problem where the involved matrix is a P-matrix and an error bound can be found by solving a linear system of equations, which is sharper than the Mathias-Pang error bound.
Abstract: We give new error bounds for the linear complementarity problem where the involved matrix is a P-matrix. Computation of rigorous error bounds can be turned into a P-matrix linear interval system. Moreover, for the involved matrix being an H-matrix with positive diagonals, an error bound can be found by solving a linear system of equations, which is sharper than the Mathias-Pang error bound. Preliminary numerical results show that the proposed error bound is efficient for verifying accuracy of approximate solutions.
142 citations
Posted Content•
TL;DR: In this article, it is argued that the procedure based on differentials is superior to other methods of differentiation, and leads inter alia to a satisfactory chain rule for matrix functions.
Abstract: Several definitions are in use for the derivative of an m × p matrix function F(X) with respect to its n × q matrix argument X. We argue that only one of these definitions is a viable one, and that to study smooth maps from the space of n × q matrices to the space of m × p matrices it is often more convenient to study the map from nq-space to mp-space. Also, several procedures exist for a calculus of functions of matrices. It is argued that the procedure based on differentials is superior to other methods of differentiation, and leads inter alia to a satisfactory chain rule for matrix functions.
115 citations
Related Topics (5)
Performance Metrics
| Year | Papers |
|---|---|
| 2022 | 1 |
| 2021 | 6 |
| 2020 | 5 |
| 2019 | 5 |
| 2018 | 5 |
| 2017 | 2 |