Results 1  10
of
38
Communication on the Grassmann Manifold: A Geometric Approach to the Noncoherent MultipleAntenna Channel
 IEEE Trans. Inform. Theory
, 2002
"... In this paper, we study the capacity of multipleantenna fading channels. We focus on the scenario where the fading coefficients vary quickly; thus an accurate estimation of the coefficients is generally not available to either the transmitter or the receiver. We use a noncoherent block fading model ..."
Abstract

Cited by 174 (5 self)
 Add to MetaCart
In this paper, we study the capacity of multipleantenna fading channels. We focus on the scenario where the fading coefficients vary quickly; thus an accurate estimation of the coefficients is generally not available to either the transmitter or the receiver. We use a noncoherent block fading model proposed by Marzetta and Hochwald. The model does not assume any channel side information at the receiver or at the transmitter, but assumes that the coefficients remain constant for a coherence interval of length symbol periods. We compute the asymptotic capacity of this channel at high signaltonoise ratio (SNR) in terms of the coherence time , the number of transmit antennas , and the number of receive antennas . While the capacity gain of the coherent multiple antenna channel is min bits per second per hertz for every 3dB increase in SNR, the corresponding gain for the noncoherent channel turns out to be (1 ) bits per second per herz, where = min 2 . The capacity expression has a geometric interpretation as sphere packing in the Grassmann manifold.
RuellePerronFrobenius Spectrum For Anosov Maps
 Nonlinearity
, 2001
"... We extend a number of results from one dimensional dynamics based on spectral properties of the RuellePerronFrobenius transfer operator to Anosov di#eomorphisms on compact manifolds. This allows to develop a direct operator approach to study ergodic properties of these maps. In particular, we show ..."
Abstract

Cited by 33 (9 self)
 Add to MetaCart
We extend a number of results from one dimensional dynamics based on spectral properties of the RuellePerronFrobenius transfer operator to Anosov di#eomorphisms on compact manifolds. This allows to develop a direct operator approach to study ergodic properties of these maps. In particular, we show that it is possible to define Banach spaces on which the transfer operator is quasicompact. (Information on the existence of an SRB measure, its smoothness properties and statistical properties readily follow from such a result.) In dimension d = 2 we show that the transfer operator associated to smooth random perturbations of the map is close, in a proper sense, to the unperturbed transfer operator. This allows to obtain easily very strong spectral stability results, which in turn imply spectral stability results for smooth deterministic perturbations as well. Finally, we are able to implement an Ulam type finite rank approximation scheme thus reducing the study of the spectral properties of the transfer operator to a finite dimensional problem. 1.
Grasp Analysis as Linear Matrix Inequality Problems
"... Three important problems in the study of grasping and manipulation by multifingered robotic hands are: (a) Given a grasp characterized by a set of contact points and the associated contact models, determine if the grasp has force closure; (b) If the grasp does not have force closure, determine if th ..."
Abstract

Cited by 33 (2 self)
 Add to MetaCart
Three important problems in the study of grasping and manipulation by multifingered robotic hands are: (a) Given a grasp characterized by a set of contact points and the associated contact models, determine if the grasp has force closure; (b) If the grasp does not have force closure, determine if the ngers are able to apply a specified resultant wrench on the object; and (c) Compute "optimal" contact forces if the answer to problem (b) is affirmative. In this paper, based on an early result by Buss, Hashimoto and Moore, which transforms the nonlinear friction cone constraints into positive definiteness of certain symmetric matrices, we further cast the friction cone constraints into linear matrix inequalities (LMIs) and formulate all three of the problems stated above as a set of convex optimization problems involving LMIs. The latter problems have been extensively studied in optimization and control community and highly efficient algorithms with polynomial time complexity are now available for their solutions. We perform simulation studies to show the simplicity and efficiency of the LMI formulation to the three problems.
Universal construction of feedback laws achieving ISS and integralISS disturbance attenuation
, 2002
"... We study nonlinear systems with both control and disturbance inputs. The main problem addressed in the paper is design of state feedback control laws that render the closedloop system integralinputto state stable (iISS) with respect to the disturbances. We introduce an appropriate concept of co ..."
Abstract

Cited by 27 (9 self)
 Add to MetaCart
We study nonlinear systems with both control and disturbance inputs. The main problem addressed in the paper is design of state feedback control laws that render the closedloop system integralinputto state stable (iISS) with respect to the disturbances. We introduce an appropriate concept of control Lyapunov function (iISSCLF), whose existence leads to an explicit construction of such a control law.
Tracking for Fully Actuated Mechanical Systems: A Geometric Framework
 AUTOMATICA
, 1997
"... We present a general framework for the control of Lagrangian systems with as many inputs as degrees of freedom. Relying on the geometry of mechanical systems on manifolds, we propose a design algorithm for the tracking problem. The notions of error function and transport map lead to a proper definit ..."
Abstract

Cited by 23 (1 self)
 Add to MetaCart
We present a general framework for the control of Lagrangian systems with as many inputs as degrees of freedom. Relying on the geometry of mechanical systems on manifolds, we propose a design algorithm for the tracking problem. The notions of error function and transport map lead to a proper definition of configuration and velocity error. These are the crucial ingredients in designing a proportional derivative feedback and feedforward controller. The proposed approach includes as special cases a variety of results on control of manipulators, pointing devices and autonomous vehicles. Our design provides particular insight into both aerospace and underwater applications where the configuration manifold is a Lie group.
Uniting Local and Global Controllers with Robustness to Vanishing Noise
 Math. Control Signals Systems
, 2000
"... We consider control systems for which we know two stabilizing controllers. One is globally asymptotically stabilizing, the other one is only locally asymptotically stabilizing but for some reason, we insist on using it in a neighborhood of the origin. We look for a uniting control law being equal ..."
Abstract

Cited by 17 (9 self)
 Add to MetaCart
We consider control systems for which we know two stabilizing controllers. One is globally asymptotically stabilizing, the other one is only locally asymptotically stabilizing but for some reason, we insist on using it in a neighborhood of the origin. We look for a uniting control law being equal to the local feedback on a neighborhood of the origin, equal to the global one outside of a larger neighborhood and being a globally stabilizing controller. We study several solutions based on continuous, discontinuous, hybrid, time varying controllers. One criterion of selection of a controller is the robustness of the stability to vanishing noise. This leads us in particular to consider a kind of generalization of Krasovskii trajectories for hybrid systems.
Tight Frames of kPlane Ridgelets and the Problem of Representing Objects Which Are Smooth Away from dDimensional Singularities in R^n
"... For each pair (n, k) with 1 ≤ k
Abstract

Cited by 15 (5 self)
 Add to MetaCart
For each pair (n, k) with 1 ≤ k<n, we construct a tight frame (ρλ: λ ∈ Λ) for L2 (Rn), which we call a frame of kplane ridgelets. The intent is to efficiently represent functions which are smooth away from singularities along kplanes in Rn. We also develop tools to help decide whether in fact kplane ridgelets provide the desired efficient representation. We first construct a waveletlike tight frame on the Xray bundle Xn,k – the fiber bundle having the Grassman manifold Gn,k of kplanes in Rn for base space, and for fibers the orthocomplements of those planes. This waveletlike tight frame is the pushout to Xn,k, via the smooth local coordinates of Gn,k, of an orthonormal basis of tensor Meyer wavelets on Euclidean space R k(n−k) × R n−k. We then use the Xray isometry [Solmon, 1976] to map this tight frame isometrically to a tight frame for L 2 (R n) – the kplane ridgelets. This construction makes analysis of a function f ∈ L2 (Rn)bykplane ridgelets identical to the analysis of the kplane Xray transform of f by an appropriate waveletlike system for Xn,k. As wavelets are typically effective at representing point singularities, it may be expected that these new systems will be effective at representing objects whose kplane Xray transform has a point singularity. Objects with discontinuities across hyperplanes are of this form, for k = n − 1.
Learning Riemannian Metrics
 In Proceedings of the 19th conference on Uncertainty in Artificial Intelligence (UAI
, 2003
"... We consider the problem of learning a Riemannian metric associated with a given differentiable manifold and a set of points. Our approach to the problem involves choosing a metric from a parametric family that is based on maximizing the inverse volume of a given dataset of points. From a stati ..."
Abstract

Cited by 13 (1 self)
 Add to MetaCart
We consider the problem of learning a Riemannian metric associated with a given differentiable manifold and a set of points. Our approach to the problem involves choosing a metric from a parametric family that is based on maximizing the inverse volume of a given dataset of points. From a statistical perspective, it is related to maximum likelihood under a model that assigns probabilities inversely proportional to the Riemannian volume element. We discuss in detail learning a metric on the multinomial simplex where the metric candidates are pullback metrics of the Fisher information under a continuous group of transformations. When applied to documents, the resulting geodesics resemble, but outperform, the TFIDF cosine similarity measure in classification.
Overlapping BlockBalanced Canonical Forms And Parametrizations: The Stable Siso Case
 In Advances in Neural Lnformation Processing Systems 6
, 1997
"... . The balanced canonical form and parametrization of Ober for the case of SISO stable systems are extended to blockbalanced canonical forms and related inputnormal forms and parametrizations. They form an overlapping atlas of parametrizations of the manifold of stable SISO systems of given order. ..."
Abstract

Cited by 7 (3 self)
 Add to MetaCart
. The balanced canonical form and parametrization of Ober for the case of SISO stable systems are extended to blockbalanced canonical forms and related inputnormal forms and parametrizations. They form an overlapping atlas of parametrizations of the manifold of stable SISO systems of given order. This extends the usefulness of these parametrizations, e.g., in gradient algorithms for system identification. As an implication of our construction it follows that each of the subsets of the parametrization of [R. Ober, Internat. J. Control, 46 (1987), pp. 643670] corresponding to a choice for the structural indices is in fact an imbedded submanifold of the manifold of stable SISO systems of fixed order. Key words. linear dynamical systems, di#erentiable manifolds, stable systems, canonical forms, atlas, system identification AMS subject classifications. 93XX, 53XX, 15XX PII. S0363012993260549 1. Introduction. In [18], [19] a canonical statespace form was presented for the set of asym...
A Geometric Approach to Blind Deconvolution with Application to Shape from Defocus
 Proc. IEEE Computer Vision and Pattern Recognition
, 2000
"... We propose a solution to the generic \bilinear calibrationestimation problem" when using a quadratic cost function and restricting to (locally) translationinvariant imaging models. We apply the solution to the problem of reconstructing the threedimensional shape and radiance of a scene from a numb ..."
Abstract

Cited by 6 (4 self)
 Add to MetaCart
We propose a solution to the generic \bilinear calibrationestimation problem" when using a quadratic cost function and restricting to (locally) translationinvariant imaging models. We apply the solution to the problem of reconstructing the threedimensional shape and radiance of a scene from a number of defocused images. Since the imaging process maps the continuum of threedimensional space onto the discrete pixel grid, rather than discretizing the continuum we exploit the structure of maps between (niteand innitedimensional) Hilbert spaces and arrive at a principled algorithm that does not involve any choice of basis or discretization. Rather, these are uniquely determined by the data, and exploited in a functional singular value decomposition in order to obtain a regularized solution. 1 Introduction An imaging system, such as the eye or a videocamera, involves a map from the threedimensional environment onto a twodimensional surface. In order to retrieve the spatial inform...