Dual Decompositions Research Articles

QR decomposition of dual matrices and its application

Dual number matrix decompositions have played an important role in fields such as kinematics and computer graphics in recent years. In this paper, we present a QR decomposition algorithm for dual number matrices. When dealing with large-scale problems, we present the thin QR decomposition of dual number matrices, along with its algorithm with column pivoting. In numerical experiments, we discuss the suitability of different QR algorithms when confronted with various large-scale dual matrices, providing their respective domains of applicability. Finally, we employ the QR decomposition of dual matrices to compute the DMPGI, attaining results of higher precision.

Anisotropic subloading surface Cam‐clay plasticity model with rotational hardening: Deformation gradient‐based formulation for finite strain

AbstractThis study is aimed at developing an anisotropic elastoplastic constitutive model for geomaterials at finite strain and its stress calculation algorithm based on the fully implicit return‐mapping scheme. The Cam‐clay plasticity model is adopted as a specific prototype model of geomaterials. As a pertinent representation of deformation‐induced anisotropy in geomaterials, nonlinear rotational hardening is incorporated into the model in a theoretically reasonable manner by introducing the dual multiplicative decompositions of the deformation gradient tensor. In addition to the usual decomposition into elastic and plastic parts, the plastic part is decomposed further into a part contributing to the rotational hardening and a remainder part. The former part leads to a back stress ratio tensor related to the rotational hardening via a hyperelastic‐type hardening rule. The constitutive theory is thereby formulated on proper intermediate configurations entirely in terms of deformation‐like tensorial variables possessing invariance property, without resort to any objective rates of stress or stress‐like variables. Combining the Cam‐clay plasticity with the concept of subloading surface, a class of unconventional plasticity, enables the model to be capable of reproducing complex hardening/softening accompanied by volumetric contractive/dilative responses. Basic characteristics and predictive capability of the proposed model, as well as the accuracy of the developed numerical scheme, are verified through several numerical examples including monotonic and cyclic loadings.

Distributed optimization via primal and dual decompositions for delay-constrained FANETs

This paper aims to optimize the different network parameters in a distributed manner for delay-constrained flying ad hoc networks (FANETs) without the global network topology information. To this end, each Unmanned Aerial Vehicle (UAV) calculates the average interference level during a certain time period to indicate the channel states. Next, we formulate the distributed optimization problem as a utility maximization problem, which jointly optimizes power control, rate allocation and delay-constrained routing. To obtain a distributed solution, a dual method is proposed to eliminate the link capacity constraint, and a primal decomposition method is employed to decouple the end-to-end delay constraint. Built on these two methods above, a distributed optimization algorithm is proposed in this work by considering the estimated one-hop delay for each transmission, which only uses the local channel information to optimize the sub-problems and limit the end-to-end delay. Finally, we deduce the relationship between the primal and dual solutions to underpin the advantage of the proposed algorithm. Experiments on simulate (and real) data demonstrate that the proposed algorithm effectively can improve network performances in terms of energy efficiency, packet timeout ratio and network throughput.

Output sensitive algorithms for approximate incidences and their applications

An ε-approximate incidence between a point and some geometric object (line, circle, plane, sphere) occurs when the point and the object lie at distance at most ε from each other. Given a set of points and a set of objects, computing the approximate incidences between them is a major step in many database and web-based applications in computer vision and graphics, including robust model fitting, approximate point pattern matching, and estimating the fundamental matrix in epipolar (stereo) geometry.In a typical approximate incidence problem of this sort, we are given a set P of m points in two or three dimensions, a set S of n objects (lines, circles, planes, spheres), and an error parameter ε>0, and our goal is to report all pairs (p,s)∈P×S that lie at distance at most ε from one another. We present efficient output-sensitive approximation algorithms for quite a few cases, including points and lines or circles in the plane, and points and planes, spheres, lines, or circles in three dimensions. Several of these cases arise in the applications mentioned above. Our algorithms report all pairs at distance ≤ε, but may also report additional pairs, all of which are guaranteed to be at distance at most αε, for some problem-dependent constant α>1. Our algorithms are based on simple primal and dual grid decompositions and are easy to implement. We note that (a) the use of duality, which leads to significant improvements in the overhead cost of the algorithms, appears to be novel for this kind of problems; (b) the correct choice of duality in some of these problems is fairly intricate and requires some care; and (c) the correctness and performance analysis of the algorithms (especially in the more advanced versions) is fairly non-trivial. We analyze our algorithms and prove guaranteed upper bounds on their running time and on the “distortion” parameter α.

Radio over Cloud (RoC): Cloud-Assisted Distributed Beamforming for Multi-Class Traffic

Cloud has yet to be applied to computationally intensive radio signal processing, due to closely coupled computing tasks resulting from interference. This paper presents a new cloud-assisted joint beamforming architecture, where computations are decoupled for individual wireless users and pipelined for cloud execution, using Difference of Convex (DC), $\ell _1$l1-norm approximations, and dual decompositions. User-specific tasks are constructed and aligned with the cloud to leverage computation reuses and minimize overhead. The time-complexity is dramatically improved to support networks with tens to hundreds of base stations and users, without compromising the sum rate and quality-of-service. Further, the superiority of DC to the state-of-the-art Weighted Minimum Mean Square Error (WMMSE) in terms of convex relaxation is observed and discussed. Corroborated by simulations, the reason is revealed as WMMSE aggressively increases the data rate at interim stages, hence adversely interacting with $\ell _1$l1-norm approximation and reducing the feasible solution regions at later stages.

Dual Low-Rank Decompositions for Robust Cross-View Learning.

Cross-view data are very popular contemporarily, as different viewpoints or sensors attempt to richly represent data in various views. However, the cross-view data from different views present a significant divergence, that is, cross-view data from the same category have a lower similarity than those in different categories but within the same view. Considering that each cross-view sample is drawn from two intertwined manifold structures, i.e., class manifold and view manifold, in this paper, we propose a robust cross-view learning framework to seek a robust view-invariant low-dimensional space. Specifically, we develop a dual low-rank decomposition technique to unweave those intertwined manifold structures from one another in the learned space. Moreover, we design two discriminative graphs to constrain the dual low-rank decompositions by fully exploring the prior knowledge. Thus, our proposed algorithm is able to capture more within-class knowledge and mitigate the view divergence to obtain a more effective view-invariant feature extractor. Furthermore, our proposed method is very flexible in addressing such a challenging cross-view learning scenario that we only obtain the view information of the training data while with the view information of the evaluation data unknown. Experiments on face and object benchmarks demonstrate the effective performance of our designed model over the state-of-the-art algorithms.

Risk contributions: duality and sensitivity

Given a decomposition of a portfolio P as a sum of K components, practitioners commonly decompose the risk of P as a corresponding sum of risk contributions. In this paper, we prove two theorems about risk contributions. The first theorem concerns a form of duality identified in Grinold [J. Portfolio Manage. 2011, 37(2), 15–30], which may be described as follows. When we view a portfolio decomposition as a coordinate representation of the portfolio with respect to a given vector-space basis, then there is a natural dual basis with respect to which there is an alternative decomposition, here referred to as the dual decomposition. The dual decomposition gives the same contributions to risk as the original decomposition. The first theorem gives necessary and sufficient conditions for a change of basis to preserve risk contributions, and shows that all such changes of basis can be explained in terms of dual decompositions. The second theorem explores sensitivity of portfolio risk to a risk regime change and indicates that large risk contributions or large risks of the components of a decomposition may be harbingers of high sensitivity. This provides a motivation for the practice of reporting both the risk contributions and the risks of the components in a decomposition.

Three Theorems on Risk Contributions

Given a decomposition P = P_1 P_2 … P_K of a portfolio P as a sum of K components, practitioners commonly decompose the risk of P as a corresponding sum of risk contributions. In this paper we prove three theorems about risk contributions. The first theorem concerns a form of duality identified in Grinold (2011), which may be described as follows. When we view a portfolio decomposition as a coordinate representation of the portfolio with respect to a given vector-space basis, then there is a natural dual basis with respect to which there is an alternative decomposition, here referred to as the dual decomposition. The dual decomposition gives the same contributions to risk as the original decomposition. The first theorem gives necessary and sufficient conditions for a change of basis to preserve risk contributions, and shows that all such changes of basis can be explained in terms of dual decompositions. The second theorem explores sensitivity of portfolio risk to a risk regime change and indicates that large risk contributions and large risks of the components P_i of a decomposition may be harbingers of high sensitivity. This provides a motivation for the practice of reporting both the risk contributions and the risks of the components in a decomposition. The third theorem provides necessary and sufficient conditions for two sets of numbers c_1,c_2,…, c_K and s_1,s_2,…, s_K to be the risk contributions and risks of some portfolio decomposition. Such risk contributions and risks can always be realized in a space of portfolios of dimension no greater than 3, a fact that places limits on what can be gleaned from knowledge of risk contributions and risks only.

Synergy and Redundancy in Dual Decompositions of Mutual Information Gain and Information Loss

Williams and Beer (2010) proposed a nonnegative mutual information decomposition, based on the construction of information gain lattices, which allows separating the information that a set of variables contains about another variable into components, interpretable as the unique information of one variable, or redundant and synergy components. In this work, we extend this framework focusing on the lattices that underpin the decomposition. We generalize the type of constructible lattices and examine the relations between different lattices, for example, relating bivariate and trivariate decompositions. We point out that, in information gain lattices, redundancy components are invariant across decompositions, but unique and synergy components are decomposition-dependent. Exploiting the connection between different lattices, we propose a procedure to construct, in the general multivariate case, information gain decompositions from measures of synergy or unique information. We then introduce an alternative type of lattices, information loss lattices, with the role and invariance properties of redundancy and synergy components reversed with respect to gain lattices, and which provide an alternative procedure to build multivariate decompositions. We finally show how information gain and information loss dual lattices lead to a self-consistent unique decomposition, which allows a deeper understanding of the origin and meaning of synergy and redundancy.

Joint Rate Maximization of Downlink and Uplink in Multiuser MIMO SWIPT Systems

Beamforming has the potential to improve the efficiency of simultaneous wireless information and power transfer (SWIPT) systems. Existing beamforming techniques have been focused on the downlink of SWIPT systems. In this paper, we optimize the beamformers and transmit duration to maximize the weighted sum rate of both the downlink and uplink in a multiuser multiple-input multiple-output (MIMO) SWIPT system. Specifically, we formulate and transform the problem into a weighted sum mean square error minimization, conduct difference of convex programming to decouple the downlink and uplink, and convert the problem to quadratic programming (QP), which can be solved iteratively in a centralized fashion. We also decentralize the QP problem using dual decompositions, and reduce the time-complexity without compromising the data rate. Moreover, our algorithms are extended to the case under imperfect channel state information. Confirmed by simulations, the proposed decentralization can dramatically reduce the time-complexity by orders of magnitude. The scalability of the proposed approach can be substantially enhanced to support medium to large networks.

Robust Multi-View Subspace Learning through Dual Low-Rank Decompositions

Multi-view data is highly common nowadays, since various view-points and different sensors tend to facilitate better data representation. However, data from different views show a large divergence. Specifically, one sample lies in two kinds of structures, one is class structure and the other is view structure, which are intertwined with one another in the original feature space. To address this, we develop a Robust Multi-view Subspace Learning algorithm (RMSL) through dual low-rank decompositions, which desires to seek a low-dimensional view-invariant subspace for multi-view data. Through dual low-rank decompositions, RMSL aims to disassemble two intertwined structures from each other in the low-dimensional subspace. Furthermore, we develop two novel graph regularizers to guide dual low-rank decompositions in a supervised fashion. In this way, the semantic gap across different views would be mitigated so that RMSL can preserve more within-class information and reduce the influence of view variance to seek a more robust low-dimensional subspace. Extensive experiments on two multi-view benchmarks, e.g., face and object images, have witnessed the superiority of our proposed algorithm, by comparing it with the state-of-the-art algorithms.

Decentralized autonomous planning of cluster reconfiguration for fractionated spacecraft

Autonomous cluster operation such as cluster reconfiguration is one of the enabling technologies for fractionated spacecraft. By virtue of the multi-agent system theory, this paper presents an organizational architecture for fractionated spacecraft, which not only enables autonomous cluster operations but also facilitates its non-traditional attributes. Within this organizational architecture, a decentralized framework is proposed to solve cluster reconfiguration problems based on primal and dual decomposition, where subgradient methods are adopted to include reconfiguration cases with non-differentiable objectives. Two typical constraints are considered: final configuration constraints representing coupling variables and collision avoidance constraints representing coupling constraints, both of which are non-convex. General schemes are proposed to convexify those constraints via the linearization and convex restriction technology. Then final configuration constraints are tackled by primal decomposition, while collision avoidance constraints by dual decomposition. To the end, multi-level primal and dual decompositions are employed to solve reconfiguration problems with both coupling variables and coupling constraints. For illustration an example of in-plane cluster reconfiguration is solved and compared with the centralized approach the solution is optimal.

To Relay or Not to Relay in Cognitive Radio Sensor Networks

A recent investigation on the medium access control (MAC) layer in cognitive radio networks (CRNs) proposed the primary packet relaying by the secondary node maintaining an extra queue used for this particular addable functionality. Nevertheless, relaying of primary packets may introduce delays on secondary packets called secondary delay and may require an additional power budget to forward the primary packets. Power budget is particularly crucial when a type of sensor network is deployed using devices of limited power resources. In this paper, admission control is employed to efficiently manage this packet-wise relaying process in cognitive radio sensor networks (CRSNs). To be specific, we assume a cognitive packet-relaying scenario with two pairs of primary and secondary users, i.e., transmitter and receiver. We analyze and formulate the secondary delay and the required power budget of the secondary sensor node in relation to the acceptance factor (i.e., admission control parameter) that indicates whether the primary packets are admitted for relaying or not. Having defined the above, we present a tradeoff between the secondary delay and the required power budget by tuning the acceptance factor, which can be tailored to specific chosen values. Based on this behavior, we formulate an optimization problem to minimize the secondary delay over the admission control parameter subject to a limit on the required power budget. Additionally, the constraints related to the stabilities of all individual queues at the primary and secondary networks are taken into account in the proposed optimization problem, due to their interdependence relations. The solution of this problem is provided using iterative decomposition methods, i.e., dual and primal decompositions, using Lagrange multipliers that simplify the original complicated problem and result in a final equivalent dual problem that includes the initial Karush–Kuhn–Tucker (KKT) conditions. We obtain the optimal acceptance factor, while in addition, we highlight the opportunities for extra delay minimization that is provided by relaxing the initial constraints through changing the values of the Lagrange multipliers. Finally, we present the behavior of the secondary delay, assuming infinite and finite queues and assessing thereby the overflow and blocking probabilities, respectively.

Dual-color decompositions at one-loop level in Yang-Mills theory

In this work we extend two dual color decompositions of Yang-Mills amplitude to one-loop level. Starting from double-copy expression we translate the Yang-Mills integrand into the dual Del Duca-Dixon-Maltoni formulation and subsequently to the dual color-ordered formula. The dual trace factors are obtained after simultaneously solving the one-loop Kleiss-Kuijf relations, the reflection relation, and the relation that expresses the kinematic factors in the dual Del Duca-Dixon-Maltoni form.

Coupled-decompositions: exploiting primal–dual interactions in convex optimization problems

Decomposition techniques implement the so-called “divide and conquer” in convex optimization problems, being primal and dual decompositions the two classical approaches. Although both solutions achieve the goal of splitting the original program into several smaller problems (called the subproblems), these techniques exhibit in general slow speed of convergence. This is a limiting factor in practice and in order to circumvent this drawback, we develop in this article the coupled-decompositions method. As a result, the number of iterations can be reduced by more than one order of magnitude. Furthermore, the new technique is self-adjustable, i.e., it does not depend on user-defined parameters, as opposite to what happens with classical strategies. Given that in signal processing applied to communications and networking we usually deal with a variety of problems that exhibit certain coupling structures, our method is useful to design decentralized as well as centralized optimization schemes with advantages over the existing techniques in the literature. In this article, we expose there different resource allocation problems where the proposed method is successfully applied.

Primal and dual-based algorithms for sensing range adjustment in WSNs

Coverage and tracking of multiple targets, are viewed as important challenges in WSNs, mainly aimed for future ubiquitous and pervasive applications. Target coverage in WSNs with large numbers of sensor nodes and targets, and with a predefined placement of sensors, may be conducted through adjusting the sensing range and considering the energy consumption related to this operation. In this paper, we encounter the problem of multiple target coverage in WSNs by determining the sensing range of each sensor node to maximize the total utility of the network. We solve this Network Utility Maximization (NUM) problem via two approaches, primal and dual decompositions, which result in iterative distributed price-based algorithms. Convergence of sensing ranges to optimal values is proved by means of stability analysis and simulation experiments. Simulation results show convergence to optimal values in few iterations, with near optimal values for the total objective function and energy consumption of nodes. These results show scalability of our algorithm, in terms of the number of iterations needed for convergence when compared with the other two methods. Furthermore, the distributed algorithm based on dual decomposition is used to cover efficiently moving targets in consecutive time intervals.

Multiple-Description $l_{1}$-Compression

Multiple descriptions (MDs) is a method to obtain reliable signal transmissions on erasure channels. An MD encoder forms several descriptions of the signal and each description is independently transmitted across an erasure channel. The reconstruction quality then depends on the set of received descriptions. In this paper, we consider the design of redundant descriptions in an MD setup using l1-minimization with Euclidean distortion constraints. In this way we are able to obtain sparse descriptions using convex optimization. The proposed method allows for an arbitrary number of descriptions and supports both symmetric and asymmetric distortion design. We show that MDs with partial overlapping information corresponds to enforcing coupled constraints in the proposed convex optimization problem. To handle the coupled constraints, we apply dual decompositions which makes first-order methods applicable and thereby admit solutions for large-scale problems, e.g., coding entire images or image sequences. We show by examples that the proposed framework generates non-trivial sparse descriptions and non-trivial refinements. We finally show that the sparse descriptions can be quantized and encoded using off-the-shell encoders such as the set partitioning in hierarchical trees (SPIHT) encoder, however, the proposed method shows a rate-distortion loss compared to state-of-the-art image MD encoders.

Delay Robustness of Nonlinear Internet Congestion Control Schemes

<para xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> Several nonlinear Internet congestion control schemes have been proposed in the previous years based on primal and dual decompositions of convex yet centralized optimization problems that describe the bandwidth allocation problem. This design methodology uses (sub)gradient methods for the design of node/link dynamics and guarantees nominal functionality for arbitrary system sizes. In this technical note we analyze the robustness of such nonlinear schemes to the presence of multiple, non-commensurate (heterogeneous) communication delays. </para>

Optimized Video Multicasting Over Wireless Ad Hoc Networks Using Distributed Algorithm

Recently there has been a compelling need to support real-time video multicast from a single source to multiple receivers in wireless ad hoc networks. The existing work uses tree-based schemes to perform video multicast. The optimization of those schemes typically requires a centralized computation, which is not suitable for wireless ad hoc networks. In this paper, we propose an optimized video multicast scheme over wireless ad hoc networks. First, we apply a prioritized coding scheme to enable the heterogeneous receivers to reconstruct the video at different quality levels. Then we formulate the video multicasting problem using the network model, the packet loss model, and the video distortion model. To solve the optimization problem, we propose a distributed algorithm to jointly optimize the source rate, the routing scheme, and the power allocation using hierarchical dual decompositions. The distributed nature of the proposed algorithm makes it very appropriate for wireless ad hoc networks. Through extensive simulations, we demonstrate that the proposed video multicast scheme can achieve much higher video quality compared to the uniform-power scheme or the tree-based routing schemes.

A large deformation theory for rate-dependent elastic–plastic materials with combined isotropic and kinematic hardening

We have developed a large deformation viscoplasticity theory with combined isotropic and kinematic hardening based on the dual decompositions F = F e F p [Kröner, E., 1960. Allgemeine kontinuumstheorie der versetzungen und eigenspannungen. Archive for Rational Mechanics and Analysis 4, 273–334] and F p = F en p F dis p [Lion, A., 2000. Constitutive modelling in finite thermoviscoplasticity: a physical approach based on nonlinear rheological models. International Journal of Plasticity 16, 469–494]. The elastic distortion F e contributes to a standard elastic free-energy ψ ( e ) , while F en p , the energetic part of F p , contributes to a defect energy ψ ( p ) – these two additive contributions to the total free energy in turn lead to the standard Cauchy stress and a back-stress. Since F e = FF p - 1 and F en p = F p F dis p - 1 , the evolution of the Cauchy stress and the back-stress in a deformation-driven problem is governed by evolution equations for F p and F dis p – the two flow rules of the theory. We have also developed a simple, stable, semi-implicit time-integration procedure for the constitutive theory for implementation in displacement-based finite element programs. The procedure that we develop is “simple” in the sense that it only involves the solution of one non-linear equation, rather than a system of non-linear equations. We show that our time-integration procedure is stable for relatively large time steps, is first-order accurate, and is objective.