Comments on "Is the Frobenius Matrix Norm Induced?" and reply

In "Is the Frobenius matrix norm induced?", the authors ask whether the Frobenius and the H/sup 2/ norms are induced. There, they claimed that the Frobenius norm is not induced and, consequently, conjectured that the H/sup 2/ norm may not be induced. In this paper, it is shown that the Frobenius norm is induced on particular matrix spaces. It is then shown that the H/sup 2/ norm is in fact induced on a particular matrix-valued L/sup /spl infin// space. A reply is provided to the comments.     

On the computation of the gap metric for LTV systems

In this paper, we consider the robust stabilization problem for linear discrete time-varying (LTV) systems using the gap metric. In particular, we show that the time-varying (TV) directed gap reduces to an operator with a TV Hankel plus Toeplitz structure. Computation of the norm of such an operator can be carried out using an iterative scheme involving a TV Hankel operator defined on a space of Hilbert–Schmidt causal operators. The “infimization” in the TV directed gap formula is shown to be, in fact, a minimum by using duality theory. The latter holds as well in the time-invariant case.     

When does the H/sup /spl infin// fixed-lag smoothing performance saturate for a finite smoothing lag?

A notable difference between the H/sup 2/ and H/sup /spl infin// smoothing is that the achievable performance in the latter problem might "saturate" as the function of the smoothing lag in the sense that there might exist a finite smoothing lag for which the achievable performance level is the same as for the infinite smoothing lag. In this note, necessary and sufficient conditions under which such a saturation occurs are derived. In particular, it is shown that the H/sup /spl infin// performance saturates only if the H/sup /spl infin// norm of the optimal error system is achieved at the infinite frequency, i.e., if the worst case disturbance for the infinite smoothing lag case can be arbitrarily fast and thus in a sense unpredictable.     

On the observability of linear differential-algebraic systems with delays

The problem of /spl Ropf//sup n/-observability is considered for the simplest linear time- delay differential-algebraic system consisting of differential and difference equations. A determining equation system is introduced and a number of algebraic properties of the determining equation solutions is established, in particular, the well-known Hamilton-Cayley matrix theorem is generalized to the solutions of determining equation. As a result, an effective parametric rank criterion for the /spl Ropf//sup n/-observability is given. A dual controllability result is also formulated.     

Minimizing the feedback matrix norm in modal control problems

In this work, we propose algorithms for optimizing the choice of feedback in the modal control problem in multidimensional linear systems with the criterion of minimizing the norm of the feedback matrix. The proposed approaches to solving this problem are based on representing the original system in an orthonormal basis. In particular, representing controllability in a block form lets us divide a high dimensional optimization problem into optimization subproblems of lower dimension. We pay special attention to finding the initial value of the feedback matrix with suboptimal search procedures. We give recommendations on constructing numerical search procedures for suboptimal solutions with gradient-based methods. The efficiency of the developed algorithms is demonstrated with numerical examples.     

基于L1/2正则化的三维人体姿态重构

针对从给定2D特征点的单目图像中重构对象的3D形状问题,本文在形状空间模型的基础上,结合L_1/2正则化和谱范数的性质提出一种基于L_1/2正则化的凸松弛方法,将形状空间模型的非凸求解问题通过凸松弛方法转化为凸规划问题;在采用ADMM算法对凸规划问题进行优化求解过程中,提出谱范数近端梯度算法保证解的正交性与稀疏性.利用所提的优化方法,基于形状空间模型和3D可变形状模型在卡内基梅隆大学运动捕获数据库上进行3D人体姿态重构,定性和定量对比实验结果表明本文方法均优于现有的优化方法,验证了所提方法的有效性.     

不确定离散系统的最优保性能控制*

对一类具有范数有界时变参数不确定性的离散时间线性系统和一个二次型性能指标,研究其最优保性能状态反馈控制律的设计问题。     

基于lP范数的非凸低秩张量最小化

在低秩矩阵、张量最小化问题中,凸函数容易求得最优解,而非凸函数可以得到更低秩的局部解.文中基于非凸替换函数的低秩张量恢复问题,提出基于l_p范数的非凸张量模型.采用迭代加权核范数算法求解模型,实现低秩张量最小化.在合成数据和真实图像上的大量实验验证文中方法的恢复性能.     

l/sub 1/-suboptimal pole placement design

We incorporate an arbitrary number of "candidate" positive distinct poles into the scalar l/sub 1/ optimization framework in order to obtain low-order rational suboptimal solutions to constrained l/sub 1/ optimization problems. Our approach uses a single linear program to minimize an upper bound on the l/sub 1/ norm and gives a solution which uses only the best out of the prespecified poles. Rational suboptimal solutions to a scalar two-block problem are obtained similarly.     

A comment on ‘ self-tuning pole/zero assignment regulators’

The generalized H ₂ optimal control problem for a linear time-invariant system is one in which the conventional H ₂ norm is replaced by an operator norm. The closed-loop system is described in terms of a mapping between the space of time-domain input disturbances in L ₂ and the space of time-domain regulated outputs in L _∞. A minimum of this norm is then sought over all stabilizing controllers. It is shown that optimal controllers for such problems have the structure of a Kalman filter with estimated state feedback, where the feedback gains are obtained from the solution to a weighted LQR problem. A computational algorithm is presented to determine the weights in this LQR problem, and examples are given which demonstrate various problems which may arise in obtaining the optimal weights. In particular, it is shown that the generalized H ₂ problem may involve the solution to a singular LQR problem.     

Projective clustering ensembles

A considerable amount of work has been done in data clustering research during the last four decades, and a myriad of methods has been proposed focusing on different data types, proximity functions, cluster representation models, and cluster presentation. However, clustering remains a challenging problem due to its ill-posed nature: it is well known that off-the-shelf clustering methods may discover different patterns in a given set of data, mainly because every clustering algorithm has its own bias resulting from the optimization of different criteria. This bias becomes even more important as in almost all real-world applications, data is inherently high-dimensional and multiple clustering solutions might be available for the same data collection. In this respect, the problems of projective clustering and clustering ensembles have been recently defined to deal with the high dimensionality and multiple clusterings issues, respectively. Nevertheless, despite such two issues can often be encountered together, existing approaches to the two problems have been developed independently of each other. In our earlier work Gullo et al. (Proceedings of the international conference on data mining (ICDM), 2009a) we introduced a novel clustering problem, called projective clustering ensembles (PCE): given a set (ensemble) of projective clustering solutions, the goal is to derive a projective consensus clustering, i.e., a projective clustering that complies with the information on object-to-cluster and the feature-to-cluster assignments given in the ensemble. In this paper, we enhance our previous study and provide theoretical and experimental insights into the PCE problem. PCE is formalized as an optimization problem and is designed to satisfy desirable requirements on independence from the specific clustering ensemble algorithm, ability to handle hard as well as soft data clustering, and different feature weightings. Two PCE formulations are defined: a two-objective optimization problem, in which the two objective functions respectively account for the object- and feature-based representations of the solutions in the ensemble, and a single-objective optimization problem, in which the object- and feature-based representations are embedded into a single function to measure the distance error between the projective consensus clustering and the projective ensemble. The significance of the proposed methods for solving the PCE problem has been shown through an extensive experimental evaluation based on several datasets and comparatively with projective clustering and clustering ensemble baselines.     

Direct state-space solution to the discrete-timeH∞ one-block problem

A general state-space representation is used to allow a complete formulation of the H_∞ optimization problem without any invertibility condition on the system matrix, unlike existing solutions. A straightforward approach is used to solve the one-block H_∞ optimization problem. The parameterization of all solutions to the discrete-time H_∞ suboptimal one-block problem is first given in transfer function form in terms of a set of functions in H _∞ that satisfy a norm bound. The parameterization of all solutions is also given as a linear fractional representation     

Numerical modeling and optimization of one-way wave processes in inhomogeneous media

The problem of numerical modeling and optimization of wave processes on the basis of a parabolic Schrodinger-type wave equation with a complex non-self-adjoint operator is considered. An optimality criterion is formulated. Properties of an extremal problem are investigated. A numerical method is proposed for the modeling and optimization of acoustic fields in inhomogeneous domains with piecewise continuous parameters.     

On the design of multiple key hashing files for concurrent orthogonal range retrieval between two disks

This paper concerns the following problem: given a set of multi-attribute records, a fixed number of buckets and a two-disk system, arrange the records into the buckets and then store the buckets between the disks in such a way that, over all possible orthogonal range queries (ORQs), the disk access concurrency is maximized. We shall adopt the multiple key hashing (MKH) method for arranging records into buckets and use the disk modulo (DM) allocation method for storing buckets onto disks. Since the DM allocation method has been shown to be superior to any other allocation methods for allocating an MKH file onto a two-disk system for answering ORQs, the real issue is knowing how to determine an optimal way for organizing the records into buckets based upon the MKH concept.

A performance formula that can be used to evaluate the average response time, over all possible ORQs, of an MKH file in a two-disk system using the DM allocation method is first presented. Based upon this formula, it is shown that our design problem is related to a notoriously difficult problem, namely the Prime Number Problem. Then a performance lower bound and an efficient algorithm for designing optimal MKH files in certain cases are presented. It is pointed out that in some cases the optimal MKH file for ORQs in a two-disk system using the DM allocation method is identical to the optimal MKH file for ORQs in a single-disk system and the optimal average response time in a two-disk system is slightly greater than one half of that in a single-disk system.     

Exact computation of traces and H/sup 2/ norms for a class of infinite-dimensional problems

We derive a formula for the trace of a class of differential operators defined by forced two point boundary value problems. The formula involves finite-dimensional computations with matrices whose dimension is no larger than the order of the differential operator. Thus, we achieve an exact reduction of an infinite-dimensional problem to a finite-dimensional one. We relate this trace calculation to computation of the H/sup 2/ norm for certain infinite-dimensional systems. An example from fluid dynamics is included to illustrate the method.     

Global Optimization through Rotation Space Search

This paper introduces a new algorithmic technique for solving certain problems in geometric computer vision. The main novelty of the method is a branch-and-bound search over rotation space, which is used in this paper to determine camera orientation. By searching over all possible rotations, problems can be reduced to known fixed-rotation problems for which optimal solutions have been previously given. In particular, a method is developed for the estimation of the essential matrix, giving the first guaranteed optimal algorithm for estimating the relative pose using a cost function based on reprojection errors. Recently convex optimization techniques have been shown to provide optimal solutions to many of the common problems in structure from motion. However, they do not apply to problems involving rotations. The search method described in this paper allows such problems to be solved optimally. Apart from the essential matrix, the algorithm is applied to the camera pose problem, providing an optimal algorithm. The approach has been implemented and tested on a number of both synthetically generated and real data sets with good performance. NICTA is funded by the Australian Government’s Backing Australia’s Ability initiative, in part through the Australian Research Council.     

On optimality of approximate low rank solutions of large-scale matrix equations

In this paper, we discuss some optimality results for the approximation of large-scale matrix equations. In particular, this includes the special case of Lyapunov and Sylvester equations, respectively. We show a relation between the iterative rational Krylov algorithm and a Riemannian optimization method which recently has been shown to locally minimize a certain energy norm of the underlying Lyapunov operator. Moreover, we extend the results for a more general setting leading to a slight modification of IRKA. By means of some numerical test examples, we show the efficiency of the proposed methods.     

Improving the Solution of Variational Inequalities Based on the Optimization Approach

An optimization approach to solving a convex finite-dimensional variational inequality with nonpotential operator is examined. It is shown how to construct an optimization problem equivalent to the variational inequality in the space of the original variables. A first-order algorithm to solve variational inequalitieis is formulated based on this optimization problem.