Constraint Subspace Research Articles

Regularized constraint subspace based method for image set classification

Subspace methods are popular for image set classification due to the excellent representation ability of subspaces. Generalized difference subspace and orthogonal subspace are two currently effective projection strategies for extracting discriminative subspaces. However, both of these methods discard part of the common subspace to form the constraint subspace, which may cause a loss of discriminative information. In this work, we combine the difference subspace and orthogonal subspace to form a full rank constraint subspace. Moreover, we generalize this approach to a common framework using eigenspectrum regularization models (ERMs). The full rank constraint subspace that is regularized by different ERMs is called the regularized constraint subspace (RCS). Furthermore, we propose a new ERM using the concept of difference subspace, namely, the difference subspace regularization model (DSRM). The DSRM and two other current ERMs are incorporated in our RCS-based framework. The results from extensive experiments have demonstrated the effectiveness of our proposed approaches.

CONTROL OF A COMPLIANT HUMANOID ROBOT IN DOUBLE SUPPORT PHASE: A GEOMETRIC APPROACH

Enhancing energy efficiency of bipedal walking is an important research problem that has been approached by design of recently developed compliant bipedal robots such as CoMan. While compliance leads to energy efficiency, it also complicates the walking control system due to further under-actuated degrees of freedom (DoF) associated with the compliant actuators. This problem becomes more challenging as the constrained motion of the robot in double support is considered. In this paper this problem is approached from a multi-variable geometric control aspect to systematically account for the compliant actuators dynamics and constrained motion of the robot in double support phase using a detailed electro-mechanical model of CoMan. It is shown that the formulation of constraint subspace is non-trivial in the case of non-rigid robots. A step-wise numerical algorithm is provided and the effectiveness of the proposed method is illustrated via simulation, using a ten DoF model of CoMan.

Fast hopfield neural networks using subspace projections

Hopfield Neural Networks are well-suited to the fast solution of complex optimization problems. Their application to real problems usually requires the satisfaction of a set of linear constraints that can be incorporated with an additional violation term. Another option proposed in the literature lies in confining the search space onto the subspace of constraints in such a way that the neuron outputs always satisfy the imposed restrictions. This paper proposes a computationally efficient subspace projection method that also includes variable updating step mechanisms. Some numerical experiments are used to verify the good performance and fast convergence of the new method.

Improving dynamic texture recognition with constraint subspace learning

Improving dynamic texture recognition with constraint subspace learning

On the Relative and Absolute Positioning Errors in Self-Localization Systems

This paper considers the accuracy of sensor node location estimates from self-calibration in sensor networks. The total parameter space is shown to have a natural decomposition into relative and centroid transformation components. A linear representation of the transformation parameter space is shown to coincide with the nullspace of the unconstrained Fisher information matrix (FIM). The centroid transformation subspace-which includes representations of rotation, translation, and scaling-is characterized for a number of measurement models including distance, time-of-arrival (TOA), time-difference-of-arrival (TDOA), angle-of-arrival (AOA), and angle-difference-of-arrival (ADOA) measurements. The error decomposition may be applied to any localization algorithm in order to better understand its performance characteristics, and it may be applied to the Cramer-Rao bound (CRB) to determine performance limits in the relative and transformation domains. A geometric interpretation of the constrained CRB is provided based on the principal angles between the measurement subspace and the constraint subspace. Examples are presented to illustrate the utility of the proposed error decomposition into relative and transformation components.

Optimal control for linear systems with state equality constraints

This paper deals with the optimal control problem for linear systems with linear state equality constraints. For deterministic linear systems, first we find various existence conditions for constraining state feedback control and determine all constraining feedback gains, from which the optimal feedback gain is derived by reducing the dimension of the control input space. For systems with stochastic noises, it is shown that the same gain used for constraining the deterministic system also optimally constrains the expectation of states inside the constraint subspace and minimizes the expectation of the squared constraint error. We compare and discuss performance differences between unconstrained (using penalty method), projected, and constrained controllers for both deterministic and stochastic systems. Finally, numerical examples are used to demonstrate the performance difference of the three controllers.

Analysis of the SQP-Method for Optimal Control Problems Governed by the Nonstationary Navier–Stokes Equations Based on $L^p$-theory

The aim of this article is to present a refined convergence theory of the SQP-method applied to optimal control problems for the nonstationary Navier–Stokes equations. We will employ a second-order sufficient optimality condition, which requires that the second derivative of the Lagrangian is positive definite on a subspace of inactive constraints. Therefore, we have to use the $L^p$-theory of optimal controls of the nonstationary Navier–Stokes equations rather than Hilbert space methods. Estimates of state and adjoint equations with respect to $L^p$-norms are provided. Finally, the local convergence of the SQP-method is confirmed by numerical tests.

OPTIMAL CONTROL OF LINEAR SYSTEMS WITH STATE EQUALITY CONSTRAINTS

This paper deals with the optimal control problem for systems with state linear equality constraints. For deterministic linear systems, first we find various existence conditions for constraining state feedback control and determine all constraining feedback gains, from which the optimal feedback gain is derived by using the result of singular optimal control. For systems with stochastic process noises, it is shown that the same gain used for constraining the deterministic system also optimally constrains the expectation of states inside the constraint subspace and minimizes the expectation of the squared constraint error. We compare performance between unconstrained and constrained controllers for both deterministic and stochastic systems.

Random projections and the optimization of an algorithm for phase retrieval

Iterative phase retrieval algorithms typically employ projections onto constraint subspaces to recover the unknown phases in the Fourier transform of an image, or, in the case of x-ray crystallography, the electron density of a molecule. For a general class of algorithms, where the basic iteration is specified by the difference map, solutions are associated with fixed points of the map, the attractive character of which determines the effectiveness of the algorithm. The behavior of the difference map near fixed points is controlled by the relative orientation of the tangent spaces of the two constraint subspaces employed by the map. Since the dimensionalities involved are always large in practical applications, it is appropriate to use random matrix theory ideas to analyze the average-case convergence at fixed points. Optimal values of the gamma parameters of the difference map are found which differ somewhat from the values previously obtained on the assumption of orthogonal tangent spaces.

Implementation of combustion chemistry by in situ adaptive tabulation of rate-controlled constrained equilibrium manifolds

A general methodology called in situ adaptive tabulation-rate-controlled constrained equilibrium (ISAT-RACCE) is developed to treat detailed chemistry in turbulent combustion calculations. This method combines dimension reduction and tabulation and can be implemented in a single computer program, which is independent of the detailed mechanism and of the level of reduction selected. The dimension reduction of a detailed mechanism is achieved by the RCCE approach, which is based on the maximum entropy principle of thermodynamics: and the tabulation is performed in situ during the combustion calculations and is made in the low-dimensional constraint subspace. This method is particularly attractive for very large kinetic mechanisms (e.g., including thousands of species) because the constrained equilibrium state depends on only a small number of constraints (or constrait potentials) which form the constraint subspace. Moreover, the unique and continuous maximum-entropy manifold determined by the RCCE assumption is well suited to the ISAT algorithm—an efficient computational technique for the implementation of combustion chemistry. Test calculations are performed for non-premixed methane/air combustion in a statistically homogeneous turbulent reactor, using a detailed kinetic mechanism with 31 species and 175 reactions. A direct approach of numerically integrating the full reaction equations (32-dimensional) is performed and the result is taken as the exact solution. The ISAT-RCCE calculations based on 10- to 16-dimensional constraint subspaces show good agreement with the accurate solution, and the accuracy of the results from the 16-dimensional ISAT-RCCE calculation is comparable to that of a reference calculation using ISAT and a 17-dimensional augmented reduced mechanism (derived from the same detailed mechanism). A speed-up factor of about 500 is obtained for the ISAT-RCCE calculation compared to the direct integration approach, which demonstrates the high efficiency of the new method.

Fictitious Domain Methods for the Numerical Solution of Two-Dimensional Scattering Problems

Fictitious domain methods for the numerical solution of two-dimensional scattering problems are considered. The original exterior boundary value problem is approximated by truncating the unbounded domain and by imposing a nonreflecting boundary condition on the artificial boundary. First-order, second-order, and exact nonreflecting boundary conditions are tested on rectangular and circular boundaries. The finite element discretizations of the corresponding approximate boundary value problems are performed using locally fitted meshes, and the discrete equations are solved with fictitious domain methods. A special finite element method using nonmatching meshes is considered. This method uses the macro-hybrid formulation based on domain decomposition to couple polar and cartesian coordinate systems. A special preconditioner based on fictitious domains is introduced for the arising algebraic saddle-point system such that the subspace of constraints becomes invariant with respect to the preconditioned iterative procedure. The performance of the new method is compared to the fictitious domain methods both with respect to accuracy and computational cost.

Les aspects fondamentaux de l'admissibilité en approximation quadratique d'applications linéaires

The quadratic approximation of a linear mapping under affine constraints is a central problem in linear statistical inference. It is based on two fundamental orderings on the set of quadratic forms defined on a finite dimensional vector space E: one is Loewner ordering, and the other based on the trace of linear mappings of E into E. The direction of the affine subspace of constraints is a tensor product of two vector spaces. The search for minimum solutions is now replaced by investigation of minimal solutions, called optimal or admissible according to the ordering. A functional approach shows that we can obtain many identical results which often only depend on the linear structure of the considered vector spaces. We prove thereby two results: The first is a significant generalization of a theorem due to A. Olsen, J. Seely, and D. Birkes [11] on the completeness of the set of optimal or admissible solutions, and the second is theorem containing some necessary conditions of admissibility. This theorem is a consequence of an extension of an important theorem due to L. R. LaMotte [9], proof of which has already been given in this general framework ([18], [19]).

Task space decomposition for invariant control of constrained robots

An approach, motivated by analytical mechanics and linear algebra methods, is proposed for task space decomposition. The approach relies on the introduction of a new set of kinematic parameters describing the constrained motion of the end-effector, using the analytical forms of material and program constraints. These parameters define a new basis in the end-effector configuration space. A general inner product characterized by the unity matrix is introduced in this basis and in its dual, which gives rise to the definition of a new set of metrics for the end-effector configuration space. The vectors defined in these bases are considered as pseudo-orthogonal. Returning to the original bases of the configuration space, the symmetric matrices for the vanishing bilinear forms can be defined. In this way, the freedom and constraint subspaces can be defined in a rigorous, analytical way. The physical meaning of the resulting metrics is explained. It is shown that the task space decomposition is invariant, although non-Euclidean metrics are being used. To illustrate the application of the methodology and to explain further the properties of the task space decomposition, two examples are presented. In the first example, the robot end-effector tracks a planar surface. A particular definition of the program constraint gives rise to the introduction of skewed bases, which better explains the inherent features of the approach. In the second example, a task of operating a planar joystick, with constrained orientation of the end-effector, is considered. The relevant bases of the task space are translated in this case and it is possible to explain other features of the method.

Second-order time-frequency synthesis of nonstationary random processes

We present time-frequency methods for the synthesis of finite-energy, nonstationary random processes. The energetic characteristics of the process to be synthesized are specified in a joint time-frequency domain via a time-frequency model function. The synthesis methods optimize the autocorrelation function of the process such that the process' Wigner-Ville spectrum is closest to the given model function. An optional signal subspace constraint allows the incorporation of additional properties such as bandlimitation and also permits the reformulation of the synthesis methods in a discrete-time setting. The synthesized process is expressed either in terms of an orthonormal basis of the constraint subspace or via its Karhunen-Loeve expansion. An example involving the prolate spheroidal functions is given, and computer simulation results are provided. >

Thermodynamics and kinetics of complex mixtures

Analysis of the chemistry and kinetics of mixtures with very large numbers of chemical compounds, such as petroleum fractions and their reaction products, can be simplified by assuming an infinite number of compounds with properties that are continuous functions of one or more variables. The limiting forms of stoichiometric, thermodynamic and kinetic concepts as the number of compounds tends to infinity is of interest for these systems. Previous treatments of this subject, such as those by De Donder (1931) and Aris and Gavalas (1966), did not exploit an important property of these systems; that the subspace of stoichiometric constraints is of finite, and in fact small, dimension even when the reaction subspace is infinite. This profoundly simplifies the geometry of the equilibrium surface in composition space. Another very useful property of kinetic functions is the detailed balance principle. Here, this is expressed in a very general form, reminiscent of the Onsager principle. The concepts are illustrated by means of concrete examples.

Thermodynamics and kinetics of complex mixtures

Analysis of the chemistry and kinetics of mixtures with very large numbers of chemical compounds, such as petroleum fractions and their reaction products, can be simplified by assuming an infinite number of compounds with properties that are continuous functions of one or more variables. The limiting forms of stoichiometric, thermodynamic and kinetic concepts as the number of compounds tends to infinity is of interest for these systems. Previous treatments of this subject, such as those by De Donder (1931) and Aris and Gavalas (1966) , did not exploit an important property of these systems; that the subspace of stoichiometric constraints is of finite, and in fact small, dimension even when the reaction subspace is infinite. This profoundly simplifies the geometry of the equilibrium surface in composition space. Another very useful property of kinetic functions is the detailed balance principle. Here, this is expressed in a very general form, reminiscent of the Onsager principle. The concepts are illustrated by means of concrete examples.

The representation problem of constrained linear elasticity

The title problem is given the following explicit solution: ℂ=ℙ D S|D, where the elasticity tensor ℂ in the constrained case is the restriction to the constraint subspace D of a corresponding unconstrained elasticity tensor S, followed by composition with the orthogonal projection ℙ D on D.

On relativistic wave equations for fermions, the causality problem for Harish-Chandra degree four, the family with one auxiliary spin

Standard relativistic wave equations for elementary particles are studied. One more step is made in a particular explicit construction of all matrices involved: parity is used to separate the constraint subspace. For the particular case in which the Harish-chandra degree of the γ algebra is four, the propagation of wave fronts is discussed. The study of a simple eigenvalue equation is suggested as a first test for acausality. The family of all equations involving only one auxiliary spin is then studied. Acausality is demonstrated for all members. Explicit construction of one member equation is carried out in order to illustrate a method by which all the relatively few members of this family can be found.

Orientation in Complementary Pivot Algorithms

Shapley introduced an index theory for bimatrix games that oriented the paths generated by the Lemke-Howson algorithm and thus partitioned equilibrium points into two sets. Here we develop a similar orientation theory for a generalized complementary pivot algorithm and apply our results to bimatrix games, the linear complementarity problem, and fixed point algorithms. We provide the weakest known general sufficiency conditions (based on those of Garcia) for Lemke's algorithm to process the linear complementarity problem. We also show that if the linear complementarity version of a quadratic programming problem is solved and gives a Kuhn-Tucker solution, then the hessian of the objective function restricted to the subspace of active constraints has positive determinant. Finally we show that fixed point algorithms will only approximate fixed points of f at which the determinant of the Jacobian of f minus the identity has the appropriate sign.