Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userRetaining positive definiteness in thresholded matrices
106
0
0.0
(
0
)
Added by
Dominique Guillot
Publication date
2011
fields
Mathematical Statistics
and research's language is
English
Ask ChatGPT about the research
No Arabic abstract
Positive definite (p.d.) matrices arise naturally in many areas within mathematics and also feature extensively in scientific applications. In modern high-dimensional applications, a common approach to finding sparse positive definite matrices is to threshold their small off-diagonal elements. This thresholding, sometimes referred to as hard-thresholding, sets small elements to zero. Thresholding has the attractive property that the resulting matrices are sparse, and are thus easier to interpret and work with. In many applications, it is often required, and thus implicitly assumed, that thresholded matrices retain positive definiteness. In this paper we formally investigate the algebraic properties of p.d. matrices which are thresholded. We demonstrate that for positive definiteness to be preserved, the pattern of elements to be set to zero has to necessarily correspond to a graph which is a union of disconnected complete components. This result rigorously demonstrates that, except in special cases, positive definiteness can be easily lost. We then proceed to demonstrate that the class of diagonally dominant matrices is not maximal in terms of retaining positive definiteness when thresholded. Consequently, we derive characterizations of matrices which retain positive definiteness when thresholded with respect to important classes of graphs. In particular, we demonstrate that retaining positive definiteness upon thresholding is governed by complex algebraic conditions.
rate research
Read More
We consider the problem of characterizing entrywise functions that preserve the cone of positive definite matrices when applied to every off-diagonal element. Our results extend theorems of Schoenberg [Duke Math. J. 9], Rudin [Duke Math. J. 26], Christensen and Ressel [Trans. Amer. Math. Soc., 243], and others, where similar problems were studied when the function is applied to all elements, including the diagonal ones. It is shown that functions that are guaranteed to preserve positive definiteness cannot at the same time induce sparsity, i.e., set elements to zero. These results have important implications for the regularization of positive definite matrices, where functions are often applied to only the off-diagonal elements to obtain sparse matrices with better properties (e.g., Markov random field/graphical model structure, better condition number). As a particular case, it is shown that emph{soft-thresholding}, a commonly used operation in modern high-dimensional probability and statistics, is not guaranteed to maintain positive definiteness, even if the original matrix is sparse. This result has a deep connection to graphs, and in particular, to the class of trees. We then proceed to fully characterize functions which do preserve positive definiteness. This characterization is in terms of absolutely monotonic functions and turns out to be quite different from the case when the function is also applied to diagonal elements. We conclude by giving bounds on the condition number of a matrix which guarantee that the regularized matrix is positive definite.
The main result of the paper gives criteria for extendibility of sesquilinear form-valued mappings defined on symmetric subsets of *-semigroups to positive definite ones. By specifying this we obtain new solutions of: * the truncated complex moment problem, * the truncated multidimensional trigonometric moment problem, * the truncated two-sided complex moment problem, as well as characterizations of unbounded subnormality and criteria for the existence of unitary power dilation.
The James-Stein estimator is an estimator of the multivariate normal mean and dominates the maximum likelihood estimator (MLE) under squared error loss. The original work inspired great interest in developing shrinkage estimators for a variety of problems. Nonetheless, research on shrinkage estimation for manifold-valued data is scarce. In this paper, we propose shrinkage estimators for the parameters of the Log-Normal distribution defined on the manifold of $N times N$ symmetric positive-definite matrices. For this manifold, we choose the Log-Euclidean metric as its Riemannian metric since it is easy to compute and is widely used in applications. By using the Log-Euclidean distance in the loss function, we derive a shrinkage estimator in an analytic form and show that it is asymptotically optimal within a large class of estimators including the MLE, which is the sample Frechet mean of the data. We demonstrate the performance of the proposed shrinkage estimator via several simulated data experiments. Furthermore, we apply the shrinkage estimator to perform statistical inference in diffusion magnetic resonance imaging problems.
We investigate eigenvectors of rank-one deformations of random matrices $boldsymbol B = boldsymbol A + theta boldsymbol {uu}^*$ in which $boldsymbol A in mathbb R^{N times N}$ is a Wigner real symmetric random matrix, $theta in mathbb R^+$, and $boldsymbol u$ is uniformly distributed on the unit sphere. It is well known that for $theta > 1$ the eigenvector associated with the largest eigenvalue of $boldsymbol B$ closely estimates $boldsymbol u$ asymptotically, while for $theta < 1$ the eigenvectors of $boldsymbol B$ are uninformative about $boldsymbol u$. We examine $mathcal O(frac{1}{N})$ correlation of eigenvectors with $boldsymbol u$ before phase transition and show that eigenvectors with larger eigenvalue exhibit stronger alignment with deforming vector through an explicit inverse law. This distribution function will be shown to be the ordinary generating function of Chebyshev polynomials of second kind. These polynomials form an orthogonal set with respect to the semicircle weighting function. This law is an increasing function in the support of semicircle law for eigenvalues $(-2: ,+2)$. Therefore, most of energy of the unknown deforming vector is concentrated in a $cN$-dimensional ($c<1$) known subspace of $boldsymbol B$. We use a combinatorial approach to prove the result.
Covariance matrix testing for high dimensional data is a fundamental problem. A large class of covariance test statistics based on certain averaged spectral statistics of the sample covariance matrix are known to obey central limit theorems under the null. However, precise understanding for the power behavior of the corresponding tests under general alternatives remains largely unknown. This paper develops a general method for analyzing the power behavior of covariance test statistics via accurate non-asymptotic power expansions. We specialize our general method to two prototypical settings of testing identity and sphericity, and derive sharp power expansion for a number of widely used tests, including the likelihood ratio tests, Ledoit-Nagao-Wolfs test, Cai-Mas test and Johns test. The power expansion for each of those tests holds uniformly over all possible alternatives under mild growth conditions on the dimension-to-sample ratio. Interestingly, although some of those tests are previously known to share the same limiting power behavior under spiked covariance alternatives with a fixed number of spikes, our new power characterizations indicate that such equivalence fails when many spikes exist. The proofs of our results combine techniques from Poincare-type inequalities, random matrices and zonal polynomials.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
