In the high-dimensional data setting, sample covariance matrix is singular. order to get a numerically stable and positive definite modification of in this paper we consider condition number constrained approximation problem present its explicit solution with respect Frobenius norm. The constraint guarantees numerical stability definiteness form simultaneously. By exploiting special structure a...

W ABSTRACT e describe a bisection method to determine the 2-norm and Frobenius norm-g distance from a given matrix A to the nearest matrix with an eigenvalue on the ima inary axis. If A is stable in the sense that its eigenvalues lie in the open left half e plane, then this distance measures how "nearly unstable" A is. Each step provides ither a rigorous upper bound or a rigorous lower bound on...

The symmetry preserving singular value decomposition (SPSVD) produces the best symmetric (low rank) approximation to a set of data. These symmetric approximations are characterized via an invariance under the action of a symmetry group on the set of data. The symmetry groups of interest consist of all the non-spherical symmetry groups in three dimensions. This set includes the rotational, refle...

We introduce an algorithm to compute tensor Interpolative Decomposition (tensor ID) for the reduction of the separation rank of Canonical Tensor Decompositions (CTDs). Tensor ID selects, for a user-defined accuracy ǫ, a near optimal subset of terms of a CTD to represent the remaining terms via a linear combination of the selected terms. Tensor ID can be used as an alternative to or in combinati...

This paper presents a method to approximate the Jacobian condition number based on computational intelligence methods by training a fuzzy system. To start with, a brief overview of the concept of robotic isotropy and the use of Jacobian norms to characterize the dexterity of a robot is presented. Then, the computational cost of the condition number is shown for both the l2-norm and the Frobeniu...

Abstract Approximating the closest positive semi-definite bisymmetric matrix using Frobenius norm to a data is important in many engineering applications, communication theory and quantum physics. In this paper, we will use interior point method solve problem. The problem be reformulated into various forms, beginning as programming later, form of mixed semidefintie second-order cone optimizatio...

We present a minimax framework for classification that considers stochastic adversarial perturbations to the training data. We show that for binary classification it is equivalent to SVM, but with a very natural interpretation of regularization parameter. In the multiclass case, we obtain that our formulation is equivalent to regularizing the hinge loss with the maximum norm of the weight vecto...

Non-negative matrix factorization (NMF) aims at finding nonnegative representations of nonnegative data. Among different NMF algorithms, alternating direction method of multipliers (ADMM) is a popular one with superior performance. However, we find that ADMM shows instability and inferior performance on real-world data like speech signals. In this paper, to solve this problem, we develop a clas...

Discussion of “Minimax Estimation of Large Covariance Matrices under L1-Norm” by Tony Cai and Harrison Zhou. To appear in Statistica Sinica. Introduction. Estimation of covariance matrices in various norms is a critical issue that finds applications in a wide range of statistical problems, and especially in principal component analysis. It is well known that, without further assumptions, the em...

Undirected graphs are often used to describe high dimensional distributions. Under sparsity conditions, the graph can be estimated using l1-penalization methods. We propose and study the following method. We combine a multiple regression approach with ideas of thresholding and refitting: first we infer a sparse undirected graphical model structure via thresholding of each among many l1-norm pen...