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We present a theory for Euclidean dimensionality reduction with subgaussian matrices which unifies several restricted isometry property and Johnson-Lindenstrauss-type results obtained earlier for specific datasets. In particular, we recover and, in several cases, improve results for sets of sparse and structured sparse vectors, low-rank matrices and tensors, and smooth manifolds. In addition, we establish a new Johnson-Lindenstrauss embedding for datasets taking the form of an infinite union of subspaces of a Hilbert space., Author(s): Sjoerd Dirksen[sup.1] Author Affiliations: (1) Hausdorff Center for Mathematics, Universität Bonn, Endenicher Allee 60, 53115, BonnGermany Introduction The analysis of high-dimensional data leads to various computational issues which are [...]

The random priority (RP) mechanism is a popular way to allocate n objects to n agents with strict ordinal preferences over the objects. In the RP mechanism, an ordering over [...]

We study a weaker formulation of the nullspace property which guarantees recovery of sparse signals from linear measurements by [l.sub.1] minimization. We require this condition to hold only with high [...]

In this paper, some properties of generalized m-power matrices are firstly obtained. Also, relative properties of such matrices are generalized to some more general cases, including the generalized m-power transformations. Keywords Generalized m-power matrix, property, generalization., [section]1. Introduction The m-idempotent matrices and m-unit-ponent matrices are two typical matrices and have many interesting properties (for example, see [1]-[6]). A matrix A [member of] [C.sup.n x n] is [...]

It is unusual for both A and [A.sup.-1] to be banded--but this can be a valuable property in applications. Block-diagonal matrices F are the simplest examples; wavelet transforms are more subtle. We show that every example can be factored into A = [F.sub.1] ... [F.sub.N] where N is controlled by the bandwidths of A and [A.sup.-1] (but not by their size, so this extends to infinite matrices and leads to new matrix groups). Bruhat | CMV matrix | factorization | wavelet | permutation doi/10.1073/pnas.1005493107

This article shows that, for both spatial lag and spatial error models with strongly connected weight matrices, maximum likelihood estimates of the spatial dependence parameter are necessarily biased downward. In [...]

The task of finding a low-rank (r) matrix that best fits an original data matrix of higher rank is a recurring problem in science and engineering. The problem becomes especially difficult when the original data matrix has some missing entries and contains an unknown additive noise term in the remaining elements. The former problem can be solved by concatenating a set of r-column matrices that share a common single r-dimensional solution space. Unfortunately, the number of possible submatrices is generally very large and, hence, the results obtained with one set of r-column matrices will generally be different from that captured by a different set. Ideally, we would like to find that solution that is least affected by noise. This requires that we determine which of the r-column matrices (i.e., which of the original feature points) are less influenced by the unknown noise term. This paper presents a criterion to successfully carry out such a selection. Our key result is to formally prove that the more distinct the r. vectors of the r-column matrices are, the less they are swayed by noise. This key result is then combined with the use of a noise model to derive an upper bound for the effect that noise and occlusions have on each of the r-column matrices. It is shown how this criterion can be effectively used to recover the noise-free matrix of rank r. Finally, we derive the affine and projective structure-from-motion (SFM) algorithms using the proposed criterion. Extensive validation on synthetic and real data sets shows the superiority of the proposed approach over the state of the art. Index Terms--Low-rank matrix, noise, missing data, random matrix, matrix perturbation, subspace analysis, structure from motion, computer vision, pattern recognition.

The two equivalent generalizations of the famous Cayley--Hamilton theorem for multivariate rational matrices over any number field are proposed, which are also generalizations of the nine known generalized Cayley--Hamilton theorems. Restricting our results to a smaller range leads to the two equivalent generalized Cayley--Hamilton theorems for multivariate polynomial matrices over any number field, one of which is more efficient in some cases than a known result. Index Terms--Cayley--Hamilton theorem, generalization for multivariate polynomial matrices, generalization for multivariate rational matrices, Laurent expansion.

This paper proposes a new class of low-density generator-matrix codes (LDGM) based on chaotic dynamical systems. The codes are designed by controlling symbolic dynamics and using linear convolutional codes. Analyzing the complex structure of chaotic systems, iterative decoding is developed. The communication performance is studied, and convergence analysis of the iterative-decoding system is presented. Finally, comparison and advantages over LDGM linear block codes in terms of encoding complexity and bit-error-rate performance are described, and possible applications of our codes are discussed. Index Terms--Chaos, encoding complexity, factor graphs, iterative-decoding analysis, low-density codes.

In the paper we consider the chance-constrained version of an affinely perturbed linear matrix inequality (LMI) constraint, assuming the primitive perturbations to be independent with light-tail distributions (e.g., bounded or [...]

In this paper, block-coded modulation is used to design a 2 x 2 multiple-input multiple-output (MIMO) space--time code for slow fading channels. The Golden Code is chosen as the inner code; the scheme is based on a set partitioning of the Golden Code using two-sided ideals whose norm is a power of two. In this case, a lower bound for the minimum determinant is given by the minimum Hamming distance. The description of the ring structure of the quotients suggests further optimization in order to improve the overall distribution of determinants. Simulation results show that the proposed schemes achieve a significant gain over the un-coded Golden Code. Index Terms--Coding gain, Golden Code, Reed--Solomon codes, space--time block codes.

In this paper, a tight upper bound on the maximum possible code size of (n, 4, 2, 1)-OOCs and some direct and recursive constructions of optimal (n, 4, 2, 1)-OOCs attaining the upper bound are given. As consequences, the following new infinite series of optimal (gn, 4, 2, 1)-OOCs are obtained: i) g [member of] {1,7,11,19,23,31,35,59,71,79,131,179,191,239,251,271, 311,359,379,419,431,439,479,491,499,571,599,631,659,719, 739,751,839,971} or g is a prime < 1000 and [equivalent to] 5 (mod 8), and n = [9.sup.h][25.sup.i][49.sup.j][p.sub.1][p.sub.2] ... [p.sub.r] where h [member of] {0, 1}, i and j are arbitrary nonnegative integers, and each [p.sub.i] is a prime [equivalent to] 1 (mod 8); ii) g = 2g' where g' [member of] {1,7,11,19,23,31,47,71,127,151,167, 191,263,271,311,359,367,383,431,439,463,479,503,631,647, 719,727, 743,823,839,863,887,911,919,967,983,991} and n = [p.sub.1][p.sub.2] ... [p.sub.r] where each [p.sub.i] is a prime [equivalent to] 1 (mod 4); iii) g [member of] {4,20} and n is any positive integer prime to 30; iv) g = 8 and n = [p.sub.1][p.sub.2] ... [p.sub.r] where each [p.sub.i] is a prime [equivalent to] 1 (mod 4) greater than 5. Index Terms--Cyclic difference family, difference matrix, optimal (n, k, [[lambda].sub.a], [[lambda].sub.c]) optical orthogonal code.

Abstract--This paper addresses the problem of blind source separation (BSS). To recover original signals, from linear instantaneous mixtures, we propose a new contrast function based on the use of a [...]

A positive-real like lemma for finite dimensional linear time invariant uncertain systems with state multiplicative noise is derived. The system uncertainties are assumed to be of either polytopic or Markov jump type. Passivity conditions for both cases are derived in terms of linear matrix inequalities. The results are used to the design a direct adaptive controller for a tracking system. Index Terms--Linear matrix inequalities (LMIs), neural network controllers (NNC), simplified adaptive control (SAC).

We consider the problem of designing distributed controllers for a class of systems which can be obtained from the interconnection of a number of identical subsystems. If the state space matrices of these systems satisfy a certain structural property, then it is possible to derive a procedure for designing a distributed controller which has the same interconnection pattern as the plant. This procedure is basically a multiobjective optimization under Linear Matrix Inequality constraints, with system norms as performance indices. The explicit expressions for computing these controllers are given for both [H.sub.[infinity]] or [H.sub.2] performance, and both for static state feedback and dynamic output feedback (in discrete time). At the end of the paper, two application examples illustrate the effectiveness of the approach. Index Terms--Decomposition, distributed control, formation flying, linear matrix inequalities (LMIs), paper machines.

This correspondence first kernalizes the region covariance matrix and formalizes the similarity metric using four block matrices. The effectiveness of the proposed methods is proven with experiments on face recognition. Index Terms--Kernalization, region covariance matrix (RCM).

The generation of sequency-ordered complex Hadamard transform (SCHT) is described that is based on the complex Rademacher matrices. The relationships between SCHT and fast Fourier transform (FFR) and unified complex Hadamard transform (UCHT) are described.

This paper discusses mathematical properties of preconditioned finite-element matrices based on vector potential formulation (A method) and vector and scalar potential formulation (A-V method) for eddy-current problems. Numerical results show that A-V method with preconditioning is stable at all frequencies in contrast to A method. In this paper, this property is mathematically discussed by considering the diagonal scaling which is one of the simple preconditioning methods. In addition, regularization of A method is discussed. Index Terms--A-V method, diagonal scaling, edge elements, finite-element method, preconditioning.

Nearly-Singular design relaxes the nonsingularity assumption of the limit weight matrix in GMM, and the nonsingularity of the limit variance matrix for the first order conditions in GEL. The sample versions of these matrices are nonsingular, but in large samples we assume these sample matrices converge to a singular matrix. This can result in size distortions for the overidentifying restrictions test and large bias for the estimators. This nearly-singular design may occur because of the similar instruments in these matrices. We derive the large sample theory for GMM and GEL estimators under nearly-singular design. The rate of convergence of the estimators is slower than root n. Keywords: Singular matrix Rate of convergence Small sample properties

Abstract For orthogonal projective matrix R, i.e., [R.sup.2] = R and [R.sup.T] = R, we say that A is generalized Hermitian matrix, if RAR = A*. In this paper, we [...]

We study analyticity, differentiability, and semismoothness of Lowner's operator and spectral functions under the framework of Euclidean Jordan algebras. In particular, we show that many optimization-related classical results in the [...]

Let M be a matroid on E ∪ {l}, where l [not member of] E is a distinguished element of M. The l-port of M is the set P = [...]

High dimensionality causes problems in various areas of statistics. A particular situation that rarely has been considered is the testing of hypotheses about multivariate regression models in which the dimension of the multivariate response is large. In this article a ridge regularization approach is proposed in which either the covariance or the correlation matrix is regularized to ensure nonsingularity irrespective of the dimensionality of the data. It is shown that the proposed approach can be derived through a penalized likelihood approach, which suggests cross-validation of the likelihood function as a natural approach for estimation of the ridge parameter. Useful properties of this likelihood estimator are derived, discussed, and demonstrated by simulation. For a class of test statistics commonly used in multivariate analysis, the proposed regularization approach is compared with some obvious alternative regularization approaches using generalized inverse and data reduction through principal components analysis. Essentially, the approaches considered differ in how they shrink eigenvalues of sample covariance and correlation matrices. This leads to predictable differences in power properties when comparing the use of different regularization approaches, as demonstrated by simulation. The proposed ridge approach has relatively good power compared with the alternatives considered. In particular, a generalized inverse is shown to perform poorly and cannot be recommended in practice. Finally, the proposed approach is used in analysis of data on macroinvertebrate biomasses that have been classified to species. KEY WORDS: Cross-validation; Generalized inverse; High-dimensional data; Multivariate analysis; Multivariate regression.

The performance of iterative decoding techniques for linear block codes correcting erasures depends very much on the sizes of the stopping sets associated with the underlying Tanner graph, or, equivalently, the parity-check matrix representing the code. In this correspondence, we introduce the notion of dead-end sets to explicitly demonstrate this dependency. The choice of the parity-check matrix entails a tradeoff between performance and complexity. We give bounds on the complexity of iterative decoders achieving optimal performance in terms of the sizes of the underlying parity-check matrices. Further, we fully characterize codes for which the optimal stopping set enumerator equals the weight enumerator. Index Terms--Dead-end set, iterative decoding, linear code, parity-check matrix, stopping set.

The enumeration and parametrization of lattices are presented for the cases where the resampling ratio is prime or a composite number.

Characterizations and computational algorithms for ongoing general gradient descent trajectories in high-dimensional measure spaces for selection of statistical model, prediction and grouping are discussed. Numerical experiments with simulated and real datasets are used to show the applicability of the algorithms.

I consider the asymptotic properties of a commonly advocated covariance matrix estimator for panel data. Under asymptotics where the cross-section dimension, n, grows large with the time dimension, T, fixed, the estimator is consistent while allowing essentially arbitrary correlation within each individual. However, many panel data sets have a non-negligible time dimension. I extend the usual analysis to cases where n and T go to infinity jointly and where T [right arrow] [infinity] with n fixed. I provide conditions under which t and F statistics based on the covariance matrix estimator provide valid inference and illustrate the properties of the estimator in a simulation study. JEL classification: C12; C13; C23 Keywords: Panel; Heteroskedasticity; Autocorrelation; Robust; Covariance matrix

The existence and construction of multimagic squares is presented by giving proof of the existence of the first 7-multimagic and the first 8-multimagic squares. These constructions are based on elementary facts from linear algebra over finite fields.

The degree of polarization is employed as a criterion to find the nine independent relations among the elements of the Mueller-Jones matrix. This procedure is applied by considering a previously determined, physically realizable Mueller matrix. On the other hand, the nine bilinear constrains are obtained by directly measuring the degree of polarization from an outgoing beam of light from an optical system by considering nine incident states of light taken from the Poincare sphere. For practical purposes, all the incident polarization states must be scanned from the Poincare sphere in order to satisfy the over-polarization and the overgain conditions, respectively, for the physical realizability of the Mueller matrix. OCIS codes: 260.0260, 260.2130, 260.5430, 120.0120, 120.2130, 120.5410.

A novel classifier is developed in a kernel feature space related to the eigenspectrum of the Laplacian data matrix. It is shown that the new classifier has performed better than the Parzen window Bayes classifier and in many cases it is comparable to the support vector machine, at a computationally lower cost.

Universally decodable matrices (UDMs) can be used for coding purposes when transmitting over slow fading channels. These matrices are parameterized by positive integers L and Y and a prime power q. The main result of this correspondence is that the simple condition L [less than or equal to] q + 1 is both necessary and sufficient for (L.N.q)-UDMs to exist. The existence proof is constructive and yields a coding scheme that is equivalent to a class of codes that was proposed by Rosenbloom and Tsfasman. Our work resolves an open problem posed recently in the literature. Index Terms--Coding for slow fading channels, full-rank condition, Rosenblonm--Tsfasman codes, universally decodable matrices (UDM).

Determinants of matrices over [Z.sub.m] are found for integers m greater than or equal to 2. Formula to find out number of matrices over Z.sub.m] when the determinant is given, is derived.

In this brief, we prove that any normal transfer matrix with nonnegative impulse response and simple real poles of second order in discrete-time linear systems admits a minimal positive realization. This result can be used to obtain a minimal positive realization of some transfer matrices of any order with simple real poles. In addition, several results obtained for single-input single-output systems are adapted to solve the positive realization problem of transfer matrices with multiple real poles in multiple-input multiple-output systems. Index Terms--Minimal realizations, multivariable systems, transfer matrices, positive realizations.

This paper investigates the problem of factor prime factorizations for n-D polynomial matrices and presents a criterion for the existence of factor prime factorizations for an important class of n-D polynomial matrices. As a by-product, we also obtain an algebraic algorithm to check n-D factor primeness in some important cases which partially solves the long-standing open problem of recognizing n-D factor prime matrices. Some problems related to the factorization methods are also studied. Several examples are given to illustrate the results. The results presented in this paper are true over any coefficient field. Index Terms--Factor prime matrix, matrix factorization, multidimensional systems, n-D polynomial matrices, polynomial ring.

For two binary codes [C.sub.1], [C.sub.2], define i([C.sub.1] [C.sub.2]) = | [C.sub.1] [intersection] [C.sub.2] | to be their intersection number. This correspondence establishes that there exist Hadamard codes of length [2.sup.t], for all t [greater than or equal to] 3, with intersection number i if and only if i [member of] {0,2,4, ... , [2t.sup.2 + 1] -12, [2.sup.t + 1]--8, [2t.sup.+ 1}. Also it is proved that for all t [greater than or equal to] 4, if there exists a Hadamard matrix of order 4.s, then there exist Hadamard codes of length [2.sup.t]+[2.sub.s] with intersection number i if and only if i [member of] {0,2,4, ... , [2.sup.t + 3]s--12, [2.sup.t + 3]s--8, [2.sup.t.+3]s}. Index Terms--Extended Hamming codes, Hadamard codes, Hadamard designs, Hadamard matrices, intersection number.

A modified matrix pencil method is proposed to estimate the frequency pairs in the signal, which has bypassed the computationally expensive pairing operation. The simulation findings reveal that the accuracy of the estimates for each frequency from this method is better than other methods and it has also provided accurate and consistent frequency estimation results that other methods are unable to provide with less or comparable computational complexity.

Decomposition methods are well-known techniques for solving quadratic programming (QP) problems arising in support vector machines (SVMs). In each iteration of a decomposition method, a small number of variables are selected and a QP problem with only the selected variables is solved. Since large matrix computations are not required, decomposition methods are applicable to large QP problems. In this paper, we will make a rigorous analysis of the global convergence of general decomposition methods for SVMs. We first introduce a relaxed version of the optimality condition for the QP problems and then prove that a decomposition method reaches a solution satisfying this relaxed optimality condition within a finite number of iterations under a very mild condition on how to select variables. Index Terms--Decomposition method, global convergence, quadratic programming (QP), support vector machines (SVMs), termination.

We consider continuous time switched systems that are stabilized via a computer. Several factors (sampling, computer computation, communications through a network, etc.) introduce model uncertainties produced by unknown varying feedback delays. These uncertainties can lead to instability when they are not taken into account. Our goal is to construct a switched digital control for continuous time switched systems that is robust to the varying feedback delay problem. The main contribution of this note is to show that the control synthesis problem in the context of unknown time varying delays can be expressed as a problem of stabilizability for uncertain systems with polytopic uncertainties. Index Terms--Linear matrix inequality (LMI), polytopic uncertainties, robustness, switched linear systems, time-varying feedback delays.

Known types of resultant matrices corresponding to one-parameter matrix polynomials are generalized to the multiparameter case. Based on the resultant approach suggested, methods for solving the following problems for multiparameter [...]

The paper presents new pseudoblock diagonal-dominance conditions for matrices with fixed block partitioning, which generalize the pointwise kth-order diagonal-dominance conditions and pointwise circuit diagonal-dominance conditions. For matrices satisfying pseudoblock conditions, [...]