Knut Hüper

Quadratically convergent algorithms for optimal dextrous hand grasping

There is a robotic balancing task, namely real-time dextrous-hand grasping, for which linearly constrained, positive definite programming gives a quite satisfactory solution from an engineering point of view. We here propose refinements of this approach to reduce the computational effort. The refinements include elimination of structural constraints in the positive definite matrices, orthogonalization of the grasp maps, and giving a precise Newton step size selection rule.

Essential Matrix Estimation Using Gauss-Newton Iterations on a Manifold

A novel approach for essential matrix estimation is presented, this being a key task in stereo vision processing. We estimate the essential matrix from point correspondences between a stereo image pair, assuming that the internal camera parameters are known. The set of essential matrices forms a smooth manifold, and a suitable cost function can be defined on this manifold such that its minimum is the desired essential matrix. We seek a computationally efficient optimization scheme towards meeting the demands of on-line processing of video images. Our work extends and improves the earlier research by Ma et al., who proposed an intrinsic Riemannian Newton method for essential matrix computations. In contrast to Ma et al., we propose three Gauss-Newton type algorithms that have improved convergence properties and reduced computational cost. The first one is based on a novel intrinsic Newton method, using the normal Riemannian metric on the manifold consisting of all essential matrices. The other two methods are Newton-like methods, that are more efficient from a numerical point of view. Local quadratic convergence of the algorithms is shown, based on a careful analysis of the underlying geometry of the problem.

Newton-like methods for numerical optimization on manifolds

Many problems in signal processing require the numerical optimization of a cost function, which is defined on a smooth manifold. Especially, orthogonally or unitarily constrained optimization problems tend to occur in signal processing tasks involving subspaces. In this paper we consider Newton-like methods for solving these types of problems. Under the assumption that the parameterization of the manifold is linked to so-called Riemannian normal coordinates our algorithms can be considered as intrinsic Newton methods. Moreover, if there is not such a relationship, we still can prove local quadratic convergence to a critical point of the cost function by means of analysis on manifolds. Our approach is demonstrated by a detailed example, i.e., computing the dominant eigenspace of a real symmetric matrix.

Gradient Flows Computing the C -numerical Range with Applications in NMR Spectroscopy

In this paper gradient flows on unitary matrices are studied that maximize the real part of the C-numerical range of an arbitrary complex n×n-matrix A. The geometry of the C-numerical range can be quite complicated and is only partially understood. A numerical discretization scheme of the gradient flow is presented that converges to the set of critical points of the cost function. Special emphasis is taken on a situation arising in NMR spectroscopy where the matrices C,A are nilpotent and the C-numerical range is a circular disk in the complex plane around the origin.

Local Convergence Analysis of FastICA and Related Algorithms

The FastICA algorithm is one of the most prominent methods to solve the problem of linear independent component analysis (ICA). Although there have been several attempts to prove local convergence properties of FastICA, rigorous analysis is still missing in the community. The major difficulty of analysis is because of the well-known sign-flipping phenomenon of FastICA, which causes the discontinuity of the corresponding FastICA map on the unit sphere. In this paper, by using the concept of principal fiber bundles, FastICA is proven to be locally quadratically convergent to a correct separation. Higher order local convergence properties of FastICA are also investigated in the framework of a scalar shift strategy. Moreover, as a parallelized version of FastICA, the so-called QR FastICA algorithm, which employs the QR decomposition (Gram-Schmidt orthonormalization process) instead of the polar decomposition, is shown to share similar local convergence properties with the original FastICA.

Analog rank filtering

The main task in rank filtering and many other nonlinear filtering operations is sorting. In this work, a nonlinear dynamical system for this operation is proposed. The sorting problem is embedded in a higher dimensional matrix-valued problem. An equivalent analog circuit consists of basic building blocks like adders, multipliers, and integrators which set up basic nonlinear processing cells. These processing cells are locally connected in a one-dimensional array of length N for a rank filter, with N input data elements taken as the initial values of the dynamical system. The time for sorting can be estimated theoretically and indicates fast convergence. In time complexity, the algorithm is of O(N). As opposed to a digital rank filter, the analog rank filter possesses a parameter to control the speed of convergence and the accuracy. >

Spin Dynamics: A Paradigm for Time Optimal Control on Compact Lie Groups

The development of efficient time optimal control strategies for coupled spin systems plays a fundamental role in nuclear magnetic resonance (NMR) spectroscopy. In particular, one of the major challenges lies in steering a given spin system to a maximum of its so-called transfer function. In this paper we study in detail these questions for a system of two weakly coupled spin-½ particles. First, we determine the set of maxima of the transfer function on the special unitary group SU(4). It is shown that the set of maxima decomposes into two connected components and an explicit description of both components is derived. Related characterizations for the restricted optimization task on the special orthogonal group SO(4) are obtained as well. In the second part, some general results on time optimal control on compact Lie groups are re-inspected. As an application of these results it is shown that each maximum of the transfer function can be reached in the same optimal time. Moreover, a global optimization algorithm is presented to explicitly construct time optimal controls for bilinear systems evolving on compact Lie groups. The algorithm is based on Lie-theoretic time optimal control results, established in [15], as well as on a recently proposed hybrid optimization method. Numerical simulations show that the algorithm performs well in the case a two spin-½ system.

Rolling Stiefel manifolds

In this article, rolling maps for real Stiefel manifolds are studied. Real Stiefel manifolds being the set of all orthonormal k-frames of an n-dimensional real Euclidean space are compact manifolds. They are considered here as rigid bodies embedded in a suitable Euclidean space such that the corresponding Euclidean group acts on the rigid body by rotations and translations in the usual way. We derive the kinematic equations describing this rolling motion.

A Jacobi-type method for computing balanced realizations

A new numerical scheme for computing balancing coordinate transformations in linear systems theory is presented. The method is closely related to the Jacobi method for diagonalizing symmetric matrices. Here the minimization of the sum of traces of the Gramians by orthogonal and nonorthogonal Jacobi-type rotations is considered. The algorithm is shown to be globally convergent to a balancing transformation that arranges the Hankel singular values in a prescribed ordering. Local quadratic convergence of the algorithm is shown.

An Intrinsic CG Algorithm for Computing Dominant Subspaces

In this paper, a conjugate gradient method on the complex Grabmann manifold is proposed that computes the k-principal components of a Hermitian (n × n)-matrix. The algorithm is at most of order O(n2k) and yields locally good convergence results.