Matrix-valued Variables Research Articles

Graph-based approximate message passing iterations

Abstract Approximate message passing (AMP) algorithms have become an important element of high-dimensional statistical inference, mostly due to their adaptability and concentration properties, the state evolution (SE) equations. This is demonstrated by the growing number of new iterations proposed for increasingly complex problems, ranging from multi-layer inference to low-rank matrix estimation with elaborate priors. In this paper, we address the following questions: is there a structure underlying all AMP iterations that unifies them in a common framework? Can we use such a structure to give a modular proof of state evolution equations, adaptable to new AMP iterations without reproducing each time the full argument? We propose an answer to both questions, showing that AMP instances can be generically indexed by an oriented graph. This enables to give a unified interpretation of these iterations, independent from the problem they solve, and a way of composing them arbitrarily. We then show that all AMP iterations indexed by such a graph verify rigorous SE equations, extending the reach of previous proofs and proving a number of recent heuristic derivations of those equations. Our proof naturally includes non-separable functions and we show how existing refinements, such as spatial coupling or matrix-valued variables, can be combined with our framework.

Elastica Models for Color Image Regularization

The choice of a proper regularization measure plays an important role in the field of image processing. One classical approach treats color images as two- dimensional surfaces embedded in a five-dimensional spatial-chromatic space. In this case, a natural regularization term arises as the image surface area. Choosing the chromatic coordinates as dominating over the spatial ones, we can think of the image spatial coordinates could as a parameterization of the image surface manifold in a three-dimensional color space. Minimizing the area of the image manifold leads to the Beltrami flow or mean curvature flow of the image surface in the three-dimensional color space, while minimizing the elastica of the image surface yields an additional interesting regularization. Recently, we proposed a color elastica model, which minimizes both the surface area and the elastica of the image manifold. In this paper, we propose to modify the color elastica and introduce two new models for color image regularization. The revised measures are motivated by the relations between the color elastica model, Euler’s elastica model, and the total variation model for gray level images. Compared to our previous color elastica model, the new models are direct extensions of Euler’s elastica model to color images. The proposed models are nonlinear and challenging to minimize. To overcome this difficulty, two operator-splitting methods are suggested. Specifically, nonlinearities are decoupled by the introduction of new vector- and matrix-valued variables. Then, the minimization problems are converted to initial value problems which are time-discretized by operator splitting. Each subproblem, after splitting, either has a closed-form solution or can be solved efficiently. The effectiveness and advantages of the proposed models are demonstrated by comprehensive experiments. The benefits of incorporating the elastica of the image surface as regularization terms compared to common alternatives are empirically validated.

A Color Elastica Model for Vector-Valued Image Regularization

Models related to the Euler's elastica energy have proven to be useful for many applications including image processing. Extending elastica models to color images and multichannel data is a challenging task, as stable and consistent numerical solvers for these geometric models often involve high order derivatives. Like the single channel Euler's elastica model and the total variation models, geometric measures that involve high order derivatives could help when considering image formation models that minimize elastic properties. In the past, the Polyakov action from high energy physics has been successfully applied to color image processing. Here, we introduce an addition to the Polyakov action for color images that minimizes the color manifold curvature. The color image curvature is computed by applying the Laplace--Beltrami operator to the color image channels. When reduced to gray-scale images, while selecting appropriate scaling between space and color, the proposed model minimizes Euler's elastica operating on the image level sets. Finding a minimizer for the proposed nonlinear geometric model is a challenge we address in this paper. Specifically, we present an operator-splitting method to minimize the proposed functional. The nonlinearity is decoupled by introducing three vector-valued and matrix-valued variables. The problem is then converted into solving for the steady state of an associated initial-value problem. The initial-value problem is time split into three fractional steps, such that each subproblem has a closed form solution, or can be solved by fast algorithms. The efficiency and robustness of the proposed method are demonstrated by systematic numerical experiments.

A Matrix-Valued Kuramoto Model

Beginning with the work of Lohe [14,15] there have been a number of papers [3,5,8,9,11] that have generalized the Kuramoto model for phase-locking to a non-commuting situation. Here we propose and analyze another such model. We consider a collection of symmetric matrix-valued variables that evolve in such a way as to try to align their eigenvector frames. The phase-locked state is one where the eigenframes all align, and thus the matrices all commute. We analyze the stability of the phase-locked state and show that it is stable. We also analyze a dynamic analog of the twist states arising in the standard Kuramoto model, and show that these twist states are dynamically unstable.

Observer-Based State Estimation of Discrete-Time Fuzzy Systems Based on a Joint Switching Mechanism for Adjacent Instants.

The problem of observer-based state estimation of discrete-time fuzzy systems is investigated by constructing a joint switching mechanism for adjacent instants. Thanks to the usage of both the proposed spatial partitioning method and a set of new free matrix-valued variables, abundant information about size differences of all the normalized fuzzy weighting functions for adjacent instants can be interactively integrated into the fuzzy observer design for the first time. Compared with the recent result from three important aspects (conservatism level, online computational burden, and offline computational burden), three positive results can be obtained: 1) it provides a chance for reducing the conservatism by a large margin; 2) the required online computational burden remains unchanged as referred ones; and 3) as a tradeoff, the required offline computational burden increases to some extent but is still affordable from the view of complexity analysis. Finally, two numerical simulations have been given to validate the effectiveness of our developed theoretical results.

Robust Fault Estimation Design for Discrete-Time Nonlinear Systems via A Modified Fuzzy Fault Estimation Observer

The paper provides relaxed designs of fault estimation observer for nonlinear dynamical plants in the Takagi–Sugeno form. Compared with previous theoretical achievements, a modified version of fuzzy fault estimation observer is implemented with the aid of the so-called maximum-priority-based switching law. Given each activated switching status, the appropriate group of designed matrices can be provided so as to explore certain key properties of the considered plants by means of introducing a set of matrix-valued variables. Owing to the reason that more abundant information of the considered plants can be updated in due course and effectively exploited for each time instant, the conservatism of the obtained result is less than previous theoretical achievements and thus the main defect of those existing methods can be overcome to some extent in practice. Finally, comparative simulation studies on the classical nonlinear truck-trailer model are given to certify the benefits of the theoretic achievement which is obtained in our study.

Fault Estimation Observer Design for Discrete-Time Takagi-Sugeno Fuzzy Systems Based on Homogenous Polynomially Parameter-Dependent Lyapunov Functions.

This paper investigates the problem of robust fault estimation (FE) observer design for discrete-time Takagi-Sugeno fuzzy systems via homogenous polynomially parameter-dependent Lyapunov functions. First, a novel framework of the fuzzy FE observer is established with the help of a maximum-minimum-priority-based switching mechanism. Then, for every activated switching case, a targeted result is achieved by the aid of exploring an important property of improved homogenous polynomials. Since the helpful information of the underlying system can be duly updated and effectively utilized at every sampled point, the conservatism of previous results is availably reduced. Furthermore, the proposed result is further improved by eliminating those redundant terms of the introduced matrix-valued variables. Simulation results based on a discrete-time nonlinear truck-trailer model are provided to show the advantages of the theoretic result that is developed in this paper.

Noncommutative extensions of elliptic integrable Euler–Arnold tops and Painlevé VI equation

In this paper we suggest generalizations of elliptic integrable tops to matrix-valued variables. Our consideration is based on the R-matrix description which provides Lax pairs in terms of quantum and classical R-matrices. First, we prove that for relativistic (and non-relativistic) tops, such Lax pairs with spectral parameters follow from the associative Yang–Baxter equation and its degenerations. Then we proceed to matrix extensions of the models and find out that some additional constraints are required for their construction. We describe a matrix version of the reduced elliptic top and verify that the latter constraints are fulfilled in this case. The construction of matrix extensions is naturally generalized to the monodromy preserving equation. In this way we get matrix extensions of the Painlevé VI equation and its multidimensional analogues written in the form of non-autonomous elliptic tops. Finally, it is mentioned that the matrix valued variables can be replaced by elements of noncommutative associative algebra. At the end of the paper we also describe special elliptic Gaudin models which can be considered as matrix extensions of the ( reduced) elliptic top.

The Sign-Definiteness Lemma and Its Applications to Robust Transceiver Optimization for Multiuser MIMO Systems

We formally generalize the sign-definiteness lemma to the case of complex-valued matrices and multiple norm-bounded uncertainties. This lemma has found many applications in the study of the stability of control systems, and in the design and optimization of robust transceivers in communications. We then present three different novel applications of this lemma in the area of multi-user multiple-input multiple-output (MIMO) robust transceiver optimization. Specifically, the scenarios of interest are: (i) robust linear beamforming in an interfering adhoc network, (ii) robust design of a general relay network, including the two-way relay channel as a special case, and (iii) a half-duplex one-way relay system with multiple relays. For these networks, we formulate the design problems of minimizing the (sum) MSE of the symbol detection subject to different average power budget constraints. We show that these design problems are non-convex (with bilinear or trilinear constraints) and semi-infinite in multiple independent uncertainty matrix-valued variables. We propose a two-stage solution where in the first step the semi-infinite constraints are converted to linear matrix inequalities using the generalized sign-definiteness lemma, and in the second step, we use an iterative algorithm based on alternating convex search (ACS). Via simulations we evaluate the performance of the proposed scheme.

Low-rank exploitation in semidefinite programming for control

Many control-related problems can be cast as semidefinite programs. Even though there exist polynomial time algorithms and excellent publicly available solvers, the time it takes to solve these problems can be excessive. What many of these problems have in common, in particular in control, is that some of the variables enter as matrix-valued variables. This leads to a low-rank structure in the basis matrices which can be exploited when forming the Newton equations. In this article, we describe how this can be done, and show how our code, called STRUL, can be used in conjunction with the semidefinite programming solver SDPT3. The idea behind the structure exploitation is classical and is implemented in LMI Lab, but we show that when using a modern semidefinite programming framework such as SDPT3, the computational time can be significantly reduced. Finally, we describe how the modelling language YALMIP has been changed in such a way that our code, which can be freely downloaded, can be interfaced using standard YALMIP commands. This greatly simplifies modelling and usage.

Linear system identification using proper orthogonal decomposition

A least-square-based method to identify the system matrices of linear dynamical systems is proposed. The primary focus is on the identification of a reduced-order model of the system operating in the mid-frequency range of vibration. Proper orthogonal decomposition (POD) is used for the model reduction. Such reduced-order model circumvents the limitations of traditional modal analysis which, although well-adapted in the low-frequency range, is prone to computational and conceptual difficulties in the mid-frequency range. The inverse problem involving the identification of the mass, damping and stiffness matrices is posed in the framework of a linear least-square estimation problem. To achieve this objective, Kronecker algebra is aptly exploited for a concise mathematical formulation to identify these matrix-valued variables. Tikhonov regularisation is used to satisfy the symmetry property of the system matrices. The application of the proposed methodology is demonstrated using an example of multiple degree-of-freedom discrete linear dynamical system. The robustness of the new methodology is investigated using a noise sensitivity study.

Optimum Linear Joint Transmit-Receive Processing for MIMO Channels with QoS Constraints

This paper considers vector communications through multiple-input multiple-output (MIMO) channels with a set of quality of service (QoS) requirements for the simultaneously established substreams. Linear transmit-receive processing (also termed linear precoder at the transmitter and linear equalizer at the receiver) is designed to satisfy the QoS constraints with minimum transmitted power (the exact conditions under which the problem becomes unfeasible are given). Although the original problem is a complicated nonconvex problem with matrix-valued variables, with the aid of majorization theory, we reformulate it as a simple convex optimization problem with scalar variables. We then propose a practical and efficient multilevel water-filling algorithm to optimally solve the problem for the general case of different QoS requirements. The optimal transmit-receive processing is shown to diagonalize the channel matrix only after a very specific prerotation of the data symbols. For situations in which the resulting transmit power is too large, we give the precise way to relax the QoS constraints in order to reduce the required power based on a perturbation analysis. We also propose a robust design under channel estimation errors that has an important interest for practical systems. Numerical results from simulations are given to support the mathematical development of the problem.