Showing papers on "Linear map published in 2014"
TL;DR: This paper characterizes the fundamental performance limits that can be expected from an ideal decoder given a general model, i.e., a general subset of simple vectors of interest, and defines instance optimality relatively to a model much beyond the traditional framework of sparse recovery.
Abstract: The primary challenge in linear inverse problems is to design stable and robust "decoders" to reconstruct high-dimensional vectors from a low-dimensional observation through a linear operator. Sparsity, low-rank, and related assumptions are typically exploited to design decoders which performance is then bounded based on some measure of deviation from the idealized model, typically using a norm. This paper focuses on characterizing the fundamental performance limits that can be expected from an ideal decoder given a general model, ie, a general subset of "simple" vectors of interest. First, we extend the so-called notion of instance optimality of a decoder to settings where one only wishes to reconstruct some part of the original high dimensional vector from a low-dimensional observation. This covers practical settings such as medical imaging of a region of interest, or audio source separation when one is only interested in estimating the contribution of a specific instrument to a musical recording. We define instance optimality relatively to a model much beyond the traditional framework of sparse recovery, and characterize the existence of an instance optimal decoder in terms of joint properties of the model and the considered linear operator. Noiseless and noise-robust settings are both considered. We show somewhat surprisingly that the existence of noise-aware instance optimal decoders for all noise levels implies the existence of a noise-blind decoder. A consequence of our results is that for models that are rich enough to contain an orthonormal basis, the existence of an L2/L2 instance optimal decoder is only possible when the linear operator is not substantially dimension-reducing. This covers well-known cases (sparse vectors, low-rank matrices) as well as a number of seemingly new situations (structured sparsity and sparse inverse covariance matrices for instance). We exhibit an operator-dependent norm which, under a model-specific generalization of the Restricted Isometry Property (RIP), always yields a feasible instance optimality and implies instance optimality property. This norm can be upper bounded by an atomic norm relative to the considered model.
98 citations
TL;DR: It is proved that testing the feasibility of IA for generic MIMO channels belongs to the bounded-error probabilistic polynomial complexity class, and an exact arithmetic version of the test is provided, which proves that the linear alignment problem is feasible when the algebraic dimension of the solution variety is larger than or equal to thedimension of the input space.
Abstract: In this paper, we consider the feasibility of linear interference alignment (IA) for multiple-input-multiple-output (MIMO) channels with constant coefficients for any number of users, antennas, and streams per user, and propose a polynomial-time test for this problem. Combining algebraic geometry techniques with differential topology ones, we first prove a result that generalizes those previously published on this topic. In particular, we consider the input set (complex projective space of MIMO interference channels), the output set (precoder and decoder Grassmannians), and the solution set (channels, decoders, and precoders satisfying the IA polynomial equations), not only as algebraic sets, but also as smooth compact manifolds. Using this mathematical framework, we prove that the linear alignment problem is feasible when the algebraic dimension of the solution variety is larger than or equal to the dimension of the input space and the linear mapping between the tangent spaces of both smooth manifolds given by the first projection is generically surjective. If that mapping is not surjective, then the solution variety projects into the input space in a singular way and the projection is a zero-measure set. This result naturally yields a simple feasibility test, which amounts to checking the rank of a matrix. We also provide an exact arithmetic version of the test, which proves that testing the feasibility of IA for generic MIMO channels belongs to the bounded-error probabilistic polynomial complexity class.
95 citations
TL;DR: In this article, a detailed treatment of subsets of essential spectra of a closed multivalued linear operator is presented, and some results on perturbation theory of the 2'×'2 matrix of multi-valued linear operators are established.
Abstract: In this paper, we investigate a detailed treatment of some subsets of essential spectra of a closed multivalued linear operator. On the following, we will establish some results on perturbation theory of 2 × 2 matrix of multivalued linear operators. Copyright © 2013 John Wiley & Sons, Ltd.
55 citations
TL;DR: In this article, the authors extend the principal component analysis (PCA) to second-order stationary vector time series in the sense that they seek for a contemporaneous linear transformation for a $p$-variate time series such that the transformed series is segmented into several lower-dimensional subseries, and those subseries are uncorrelated with each other both contemporaneously and serially.
Abstract: We extend the principal component analysis (PCA) to second-order stationary vector time series in the sense that we seek for a contemporaneous linear transformation for a $p$-variate time series such that the transformed series is segmented into several lower-dimensional subseries, and those subseries are uncorrelated with each other both contemporaneously and serially. Therefore those lower-dimensional series can be analysed separately as far as the linear dynamic structure is concerned. Technically it boils down to an eigenanalysis for a positive definite matrix. When $p$ is large, an additional step is required to perform a permutation in terms of either maximum cross-correlations or FDR based on multiple tests. The asymptotic theory is established for both fixed $p$ and diverging $p$ when the sample size $n$ tends to infinity. Numerical experiments with both simulated and real data sets indicate that the proposed method is an effective initial step in analysing multiple time series data, which leads to substantial dimension reduction in modelling and forecasting high-dimensional linear dynamical structures. Unlike PCA for independent data, there is no guarantee that the required linear transformation exists. When it does not, the proposed method provides an approximate segmentation which leads to the advantages in, for example, forecasting for future values. The method can also be adapted to segment multiple volatility processes.
53 citations
TL;DR: In this article, the concept of demicompactness with respect to a closed densely defined linear operator is introduced, and the relative Schechter's and approximate essential spectrum is characterized.
41 citations
TL;DR: This work establishes basic facts about the varieties of homogeneous polynomials divisible by powers of linear forms, and proves asymptotic injectivity of the Foulkes–Howe map.
Abstract: We establish basic facts about the varieties of homogeneous polynomials divisible by powers of linear forms, and explain consequences for geometric complexity theory. This includes quadratic set-theoretic equations, a description of the ideal in terms of the kernel of a linear map that generalizes the Foulkes–Howe map, and an explicit description of the coordinate ring of the normalization. We also prove asymptotic injectivity of the Foulkes–Howe map.
40 citations
Posted Content•
TL;DR: The convergence of compressive sensing based sampling techniques for the efficient evaluation of functionals of solutions for a class of high-dimensional, affine-parametric, linear operator equations which depend on possibly infinitely many parameters is analyzed.
Abstract: We analyze the convergence of compressive sensing based sampling techniques for the efficient evaluation of functionals of solutions for a class of high-dimensional, affine-parametric, linear operator equations which depend on possibly infinitely many parameters. The proposed algorithms are based on so-called "non-intrusive" sampling of the high-dimensional parameter space, reminiscent of Monte-Carlo sampling. In contrast to Monte-Carlo, however, a functional of the parametric solution is then computed via compressive sensing methods from samples of functionals of the solution. A key ingredient in our analysis of independent interest consists in a generalization of recent results on the approximate sparsity of generalized polynomial chaos representations (gpc) of the parametric solution families, in terms of the gpc series with respect to tensorized Chebyshev polynomials. In particular, we establish sufficient conditions on the parametric inputs to the parametric operator equation such that the Chebyshev coefficients of the gpc expansion are contained in certain weighted $\ell_p$-spaces for $0
39 citations
TL;DR: In this article, the authors consider parabolic equations with mixed boundary conditions and domain inhomogeneities supported on a lower dimensional hypersurface, enforcing a jump in the conormal derivative.
Abstract: We consider parabolic equations with mixed boundary conditions and domain inhomogeneities supported on a lower dimensional hypersurface, enforcing a jump in the conormal derivative. Only minimal regularity assumptions on the domain and the coefficients are imposed. It is shown that the corresponding linear operator enjoys maximal parabolic regularity in a suitable $$L^p$$
-setting. The linear results suffice to treat also the corresponding nondegenerate quasilinear problems.
37 citations
TL;DR: In this paper, a linear transformation is proposed to deal with the consensus problem of high-order linear multi-agent systems (LMASs) under a generalized linear protocol, and a necessary and sufficient criterion of consensus convergence in terms of Hurwitz stability of a matrix and an analytical expression of the consensus function are provided.
37 citations
TL;DR: This article gives a comprehensive tutorial on abstract acceleration: its origins in Presburger-based acceleration including new insights w.r.t. the linear accelerability of linear transformations, methods for simple and nested loops, recent extensions, tools and applications, and a detailed discussion of related methods and future perspectives.
33 citations
TL;DR: In this paper, a linear continuous-time infinite-dimensional system is described by a closed, densely defined linear operator that generates a continuous semigroup of bounded operators on the Hilbert space of states.
Abstract: This paper is focused on adaptively controlling a linear infinite-dimensional system to track a finite-dimensional reference model. Given a linear continuous-time infinite-dimensional plant on a Hilbert space with disturbances of known waveform but unknown amplitude and phase, we show that there exists a stabilizing direct model reference adaptive control law with the properties of certain disturbance rejection and robustness. The plant is described by a closed, densely defined linear operator that generates a continuous semigroup of bounded operators on the Hilbert space of states. The central result will show that all errors will converge to a prescribed neighborhood of zero in an infinite-dimensional Hilbert space. The result will not require the use of the standard Barbalat's lemma which requires certain signals to be uniformly continuous. This result is used to determine conditions under which a linear infinite-dimensional system can be directly adaptively controlled to follow a reference model. In particular, we examine conditions for a set of ideal trajectories to exist for the tracking problem. Our results are applied to adaptive control of general linear diffusion systems described by self-adjoint operators with compact resolvent.
TL;DR: The main assumption is that the primal cone is formed as a direct product of many small-dimensional convex cones and that the matrix of the corresponding linear operator is uniformly sparse, which can approximate the primal-dual optimal solution with accuracy $O\big({1 \over \epsilon^2}\big)$ iterations.
Abstract: In this paper, we develop a primal-dual subgradient method for solving huge-scale linear conic optimization problems. Our main assumption is that the primal cone is formed as a direct product of many small-dimensional convex cones and that the matrix $A$ of the corresponding linear operator is uniformly sparse. In this case, our method can approximate the primal-dual optimal solution with accuracy $\epsilon$ in $O\big({1 \over \epsilon^2}\big)$ iterations. At the same time, the complexity of each iteration of this scheme does not exceed $O(r q \log_2 n)$ operations, where $r$ and $q$ are the maximal numbers of nonzero elements in the rows and columns of matrix $A$, and $n$ is the number of variables.
Posted Content•
TL;DR: In this article, the authors use the Minlos-Bochner theorem to prove the existence of a broad class of sparser stochastic processes, called innovation processes, which are defined as linear transformations of general continuous-domain white noises.
Abstract: The characteristic functional is the infinite-dimensional generalization of the Fourier transform for measures on function spaces. It characterizes the statistical law of the associated stochastic process in the same way as a characteristic function specifies the probability distribution of its corresponding random variable. Our goal in this work is to lay the foundations of the innovation model, a (possibly) non-Gaussian probabilistic model for sparse signals. This is achieved by using the characteristic functional to specify sparse stochastic processes that are defined as linear transformations of general continuous-domain white noises (also called innovation processes). We prove the existence of a broad class of sparse processes by using the Minlos-Bochner theorem. This requires a careful study of the regularity properties, especially the boundedness in Lp-spaces, of the characteristic functional of the innovations. We are especially interested in the functionals that are only defined for p<1 since they appear to be associated with the sparser kind of processes. Finally, we apply our main theorem of existence to two specific subclasses of processes with specific invariance properties.
TL;DR: The exponential stability analysis for ODE switched systems with time delay is extended to distributed parameter switched systems (DPSS) in Hilbert space by applying piecewise Lyapunov-Krasovskii functionals method incorporated average dwell time approach to derive sufficient conditions for exponential stability.
Abstract: In this paper, the exponential stability analysis for ODE switched systems with time delay is extended to distributed parameter switched systems (DPSS) in Hilbert space. For a given family of exponential stable subsystems, this paper focuses on finding conditions to guarantee the overall DPSS’ exponential stability. Based on semigroup theory, by applying piecewise Lyapunov-Krasovskii functionals method incorporated average dwell time approach, sufficient conditions for exponential stability are derived. These conditions are given in the form of linear operator inequalities (LOIs) where the decision variables are operators in Hilbert space, and the stability properties depend on switching rule. Being applied to heat switched propagation equations, these LOIs are reduced to standard Linear Matrix Inequalities (LMIs). Finally, a numerical example is given to illustrate the effectiveness of the proposed result.
TL;DR: A class of Bregman divergences called the (ρ,τ) -divergence, which generates a dually flat structure in the manifold of positive measures, as well as in the manifolds of positive-definite matrices, requires the invariance under linear transformations.
Abstract: Information geometry studies the dually flat structure of a manifold, highlighted by the generalized Pythagorean theorem. The present paper studies a class of Bregman divergences called the (ρ,τ)-divergence. A (ρ,τ) -divergence generates a dually flat structure in the manifold of positive measures, as well as in the manifold of positive-definite matrices. The class is composed of decomposable divergences, which are written as a sum of componentwise divergences. Conversely, a decomposable dually flat divergence is shown to be a (ρ,τ) -divergence. A (ρ,τ) -divergence is determined from two monotone scalar functions, ρ and τ. The class includes the KL-divergence, α-, β- and (α, β)-divergences as special cases. The transformation between an affine parameter and its dual is easily calculated in the case of a decomposable divergence. Therefore, such a divergence is useful for obtaining the center for a cluster of points, which will be applied to classification and information retrieval in vision. For the manifold of positive-definite matrices, in addition to the dually flatness and decomposability, we require the invariance under linear transformations, in particular under orthogonal transformations. This opens a way to define a new class of divergences, called the (ρ,τ) -structure in the manifold of positive-definite matrices.
Posted Content•
[...]
TL;DR: In this article, it was shown that a regular hom-Lie algebra structure on a vector space can be characterized by Dirac structures in the corresponding omni-hom-Lie algebras.
Abstract: In this paper, first we show that $(\g,[\cdot,\cdot],\alpha)$ is a hom-Lie algebra if and only if $(\Lambda \g^*,\alpha^*,d)$ is an $(\alpha^*,\alpha^*)$-differential graded commutative algebra. Then, we revisit representations of hom-Lie algebras, and show that there are a series of coboundary operators. We also introduce the notion of an omni-hom-Lie algebra associated to a vector space and an invertible linear map. We show that regular hom-Lie algebra structures on a vector space can be characterized by Dirac structures in the corresponding omni-hom-Lie algebra. The underlying algebraic structure of the omni-hom-Lie algebra is a hom-Leibniz algebra, or a hom-Lie 2-algebra.
TL;DR: In this paper, a successive projection algorithm for solving the multiple-sets split feasibility problem is presented, in which the step size is directly computed, which is not needed to compute the largest eigenvalue of the related matrix or estimate the Lipschitz constant, which usually requires many inner iterations.
Abstract: The multiple-sets split feasibility problem (MSFP) arises in many areas and it can be unified as a model for many inverse problems where the constraints are required on the solutions in the domain of a linear operator as well as in the operator's range. Some existing algorithms, in order to get the suitable step size, need to compute the largest eigenvalue of the related matrix, estimate the Lipschitz constant, or use some step-size search scheme, which usually requires many inner iterations. In this article, we introduce a successive projection algorithm for solving the multiple-sets split feasibility problem. In each iteration of this algorithm, the step size is directly computed, which is not needed to compute the largest eigenvalue of the matrix or estimate the Lipschitz constant. It also does not need any step-size search scheme. Its theoretical convergence results are also given.
TL;DR: In this article, the authors investigated star-likeness condi- tions for a differential operator by using the concept of quantum calculus in the unit disk and showed that quantum calculus can be used to obtain star-like properties for the differential operator.
Abstract: In the present paper, we investigate starlikeness condi- tions for $ q$ − differential operator by using the concept of quantum calculus in the unit disk.
TL;DR: This work considers multiple auxiliary linear operators, in order to find the best operator which permits accuracy after relatively few terms are calculated, and selects the convergence control parameter optimally through the construction of an optimal control problem for the minimization of the accumulated L2-norm of the residual errors.
Abstract: Analytical solutions for the Cahn-Hilliard initial value problem are obtained through an application of the homotopy analysis method. While there exist numerical results in the literature for the Cahn-Hilliard equation, a nonlinear partial differential equation, the present results are completely analytical. In order to obtain accurate approximate analytical solutions, we consider multiple auxiliary linear operators, in order to find the best operator which permits accuracy after relatively few terms are calculated. We also select the convergence control parameter optimally, through the construction of an optimal control problem for the minimization of the accumulated L 2-norm of the residual errors. In this way, we obtain optimal homotopy analysis solutions for this complicated nonlinear initial value problem. A variety of initial conditions are selected, in order to fully demonstrate the range of solutions possible.
TL;DR: A system of regularized complete linear discriminant analysis (RCLDA) is developed that can be directly applied to small sample size (SSS) problems in which the number of available training samples is smaller than the dimensionality of the sample space.
4 May 2014
TL;DR: This work addresses a problem of industrial significance: decomposition of mixed-pixels in Time-of-Flight/Range imaging and derives Cramér-Rao Bounds for performance characterization.
Abstract: In this paper, we consider the problem of Sparse Linear Operator identification which is also linked with the topic of Sparse Deconvolution. In its abstract form, the problem can be stated as follows: Given a well behaved probing function, is it possible to identify a Sparse Linear Operator from its response to the function? We present a constructive solution to this problem. Furthermore, our approach is devoid of any sparsity inducing penalty term and explores the idea of parametric modeling. Consequently, our algorithm is non-iterative by design and circumvents tuning of any regularization parameter. Our approach is computationally efficient when compared the l
0
/l
1
-norm regularized counterparts. Our work addresses a problem of industrial significance: decomposition of mixed-pixels in Time-of-Flight/Range imaging. In this case, each pixel records range measurements from multiple contributing depths and the goal is to isolate each depth. Practical experiments corroborate our theoretical set-up and establish the efficiency of our approach, that is, speed-up in processing with lesser mean squared error. We also derive Cramer-Rao Bounds for performance characterization.
TL;DR: In this paper, the authors obtained the approximate controllability of a parabolic integrodifferential equation with interior control by using the linear operator semigroup theory and the control theory for abstract equations.
Abstract: In this paper, we obtain the approximate controllability of a parabolic integrodifferential equation with interior controls. The proof relies on the linear operator semigroup theory and the controllability theory for abstract equations. Copyright © 2013 John Wiley & Sons, Ltd.
TL;DR: In this paper, the authors studied dynamic minimization problems of the calculus of variations with generalized Lagrangian functionals that depend on a general linear operator $K$ and are defined on bounded-time intervals.
Abstract: We study dynamic minimization problems of the calculus of variations with generalized Lagrangian functionals that depend on a general linear operator $K$ and are defined on bounded-time intervals. Under assumptions of regularity, convexity and coercivity, we derive sufficient conditions ensuring the existence of a minimizer. Finally, we obtain necessary optimality conditions of Euler-Lagrange type. The main results are illustrated with special cases, when $K$ is a general kernel operator and, in particular, with $K$ being the fractional integral of Riemann-Liouville and Hadamard. The application of our results to the recent fractional calculus of variations gives answer to an open question posed in [Abstr. Appl. Anal. 2012, Art. ID 871912; doi:10.1155/2012/871912].
TL;DR: This paper shows a method to find a linear transformation that reduces the number of variables to represent a given incompletely specified index generation function and first generates the difference matrix, and then finds a minimal set of variables using a covering table.
Abstract: This paper shows a method to find a linear transformation that reduces the number of variables to represent a given incompletely specified index generation function. It first generates the difference matrix, and then finds a minimal set of variables using a covering table. Linear transformations are used to modify the covering table to produce a smaller solution. Reduction of the difference matrix is also considered. key words: minimal cover, linear transformation, functional decomposition, incompletely specified function, logic minimization
TL;DR: This article deals with linear prediction in large dimensions by obtaining various explicit forms of the best linear predictor in a Hilbert space from the fact that the associated linear operator is, in general, not continuous.
TL;DR: This work presents abstract acceleration techniques for computing loop invariants for numerical programs with linear assignments and conditionals and shows how these techniques can be adapted to solve the challenge of linear programming.
Abstract: We present abstract acceleration techniques for computing loop invariants for numerical programs with linear assignments and conditionals. Whereas abstract interpretation techniques typically over-...
TL;DR: In this paper, it was shown that a linear map satisfies for all and if and only if there is a unitary such that one of the following holds: identity map or the transposition map.
Abstract: For a positive integer , let be the set of complex matrices. Suppose are positive integers and . Denote by the -numerical range of a matrix . It is shown that a linear map satisfiesfor all and if and only if there is a unitary such that one of the following holds. For all , , and for all , , where (1) is the identity map or the transposition map , or (2) and has the form or .
TL;DR: A simple class of variational integrators for linear second order mechanical systems are discussed and a constrained identification technique is proposed which employs simple linear transformation formulas to recover the continuous time parameters of the system from the discrete-time identified model.
1 Jan 2014
TL;DR: In this paper, the essentially semi-regular spectrum of closed, densely defined linear operator is stable under commuting compact perturbation and its Kato spectrum is stable subjected to additive commuting nilpotent perturbations.
Abstract: In this paper, we give some proprieties of the semi-regular, essentially semi-regular and the operators of Kato type on a Banach space We also show that the essentially semi-regular spectrum of closed, densely defined linear operator is stable under commuting compact perturbation and its Kato spectrum is stable subjected to additive commuting nilpotent perturbations
Posted Content•
TL;DR: In this paper, the authors studied dynamic minimization problems of the calculus of variations with generalized Lagrangian functionals that depend on a general linear operator $K$ and defined on bounded-time intervals.
Abstract: We study dynamic minimization problems of the calculus of variations with generalized Lagrangian functionals that depend on a general linear operator $K$ and defined on bounded-time intervals. Under assumptions of regularity, convexity and coercivity, we derive sufficient conditions ensuring the existence of a minimizer. Finally, we obtain necessary optimality conditions of Euler-Lagrange type. Main results are illustrated with special cases, when $K$ is a general kernel operator and, in particular, with $K$ the fractional integral of Riemann-Liouville and Hadamard. The application of our results to the recent fractional calculus of variations gives answer to an open question posed in [Abstr. Appl. Anal. 2012, Art. ID 871912; doi:10.1155/2012/871912].