Abstract-The squared distance function is one of the standard functions on which an optimization algorithm is commonly run, whether it is used directly or chained with other functions. Illustrative examples include center of mass computation, implementation of k-means algorithm and robot positioning. This function can have a simple expression (as in the Euclidean case), or it might not even have a closed form expression. Nonetheless, when used in an optimization problem formulated on non-Euclidean manifolds, the appropriate (intrinsic) version must be used and depending on the algorithm, its gradient and/or Hessian must be computed. For many commonly used manifolds a way to compute the intrinsic distance is available as well as its gradient, the Hessian however is usually a much more involved process, rendering Newton methods unusable on many standard manifolds. This article presents a way of computing the Hessian on connected locally-symmetric spaces on which standard Riemannian operations are known (exponential map, logarithm map and curvature). Although not a requirement for the result, describing the manifold as naturally reductive homogeneous spaces, a special class of manifolds, provides a way of computing these functions. The main example focused in this article is centroid computation of a finite constellation of points on connected locally symmetric manifolds since it is directly formulated as an intrinsic squared distance optimization problem. Simulation results shown here confirm the quadratic convergence rate of a Newton algorithm on commonly used manifolds such as the sphere, special orthogonal group, special Euclidean group, symmetric positive definite matrices, Grassmann manifold and projective space.
We address the problem of space-time codebook design for non-coherent communications in multiple-antenna wireless systems. In contrast with other approaches, the channel matrix is modeled as an unknown deterministic parameter at both the receiver and the transmitter, and the Gaussian observation noise is allowed to have an arbitrary correlation structure, known by the transmitter and the receiver. In order to handle the unknown deterministic spacetime channel, a generalized likelihood ratio test (GLRT) receiver is considered. A new methodology for space-time codebook design under this non-coherent setup is proposed. This optimizes the probability of error of the GLRT receiver's detector in the high signal-to-noise ratio (SNR) regime, thus solving a high-dimensional nonlinear nonsmooth optimization problem in a two-step approach: (i) firstly, a convex SDP relaxation of the codebook design problem yields a rough estimate of the optimal codebook; (ii) this is then refined through a geodesic descent optimization algorithm that exploits the Riemannian geometry imposed by the power constraints on the spacetime codewords. The results obtained through computer simulations illustrate the advantages of our method. For the specific case of spatio-temporal white observation noise, our codebook constructions replicate the performance of state-of-art known solutions. The main point here is that our methodology permits to extend the codebook construction to any given correlated noise environment. The simulation results show the good performance of these new designed codes.
Index TermsMultiple-input multiple output (MIMO) systems, non-coherent communications, space-time constellations, generalized likelihood ratio test (GLRT) receiver, semidefinite programming (SDP), geodesic descent algorithm (GDA).
We consider statistical models parameterized over connected Rie mannian manifolds. We present a lower bound on the mean-square distance of unbiased estimators about their mean values. The de rived bound depends both on the curvature of the parameter man ifold and a coordinate-free extension of the classical Fisher in formation matrix. Our study can be applied in estimation prob lems with smooth parametric constraints, and in statistical models indexed over coset spaces. Illustrative examples concerning in ferences on the unit-sphere and the complex projective space are worked out
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