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Computational models for the neural control of movement must take into account the properties of sensorimotor systems, including the signal-dependent intensity of the noise and the transmission delay affecting the signal conduction. For this purpose, this paper presents an algorithm for model-based control and estimation of a class of linear stochastic systems subject to multiplicative noise affecting the control and feedback signals. The state estimator based on Kalman filtering is allowed to take into account the current feedback to compute the current state estimate. The optimal feedback control process is adapted accordingly. The resulting estimation error is smaller than the estimation error obtained when the current state must be predicted based on the last feedback signal, which reduces variability of the simulated trajectories. In particular, the performance of the present algorithm is good in a range of feedback delay that is compatible with the delay induced by the neural transmission of the sensory inflow.

Many problems in the sciences and engineering can be rephrased as optimization problems on matrix search spaces endowed with a so-called manifold structure. This book shows how to exploit the special structure of such problems to develop efficient numeric

The classes of continuous-time flows on Rn×p that induce the same flow on the set of p- dimensional subspaces of Rn×p are described. The power flow is briefly reviewed in this framework, and a subspace generalization of the Rayleigh quotient flow [Linea

Many problems in the sciences and engineering can be rephrased as optimization problems on matrix search spaces endowed with a so-called manifold structure. This book shows how to exploit the special structure of such problems to develop efficient numerical algorithms. It places careful emphasis on both the numerical formulation of the algorithm and its differential geometric abstraction--illustrating how good algorithms draw equally from the insights of differential geometry, optimization, and numerical analysis. Two more theoretical chapters provide readers with the background in differential geometry necessary to algorithmic development. In the other chapters, several well-known optimization methods such as steepest descent and conjugate gradients are generalized to abstract manifolds. The book provides a generic development of each of these methods, building upon the material of the geometric chapters. It then guides readers through the calculations that turn these geometrically formulated methods into concrete numerical algorithms. The state-of-the-art algorithms given as examples are competitive with the best existing algorithms for a selection of eigenspace problems in numerical linear algebra. Optimization Algorithms on Matrix Manifolds offers techniques with broad applications in linear algebra, signal processing, data mining, computer vision, and statistical analysis. It can serve as a graduate-level textbook and will be of interest to applied mathematicians, engineers, and computer scientists.

We propose a Newton-like iteration that evolves on the set of fixed dimensional subspaces of ℝn and converges locally cubically to the invariant subspaces of a symmetric matrix. This iteration is compared in terms of numerical cost and global behavior w

We give simple formulas for the canonical metric, gradient, Lie derivative, Riemannian connection, parallel translation, geodesics and distance on the Grassmann manifold of p-planes in ℝn. In these formulas, p-planes are represented as the column space

We study the global behaviour of a Newton algorithm on the Grassmann manifold for invariant subspace computation. It is shown that the basins of attraction of the invariant subspaces may collapse in case of small eigenvalue gaps. A Levenberg-Marquardt-like modification of the algorithm with low numerical cost is proposed. A simple strategy for choosing the parameter is shown to dramatically enlarge the basins of attraction of the invariant subspaces while preserving the fast local convergence.

The classical Rayleigh quotient iteration (RQI) allows one to compute a one-dimensional invariant subspace of a symmetric matrix A. Here we propose a generalization of the RQI which computes a p-dimensional invariant subspace of A. Cubic convergence is preserved and the cost per iteration is low compared to other methods proposed in the literature.

The classical Rayleigh quotient iteration (RQI) allows one to compute a one-dimensional invariant subspace of a symmetric matrix A. Here we propose a generalization of the RQI which computes a p-dimensional invariant subspace of A. Cubic convergence is preserved and the cost per iteration is low compared to other methods proposed in the literature.