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High order smoothing splines versus least squares problems on Riemannian manifolds

In this paper, we present a generalization of the classical least squares problem on Euclidean spaces, introduced by Lagrange, to more general Riemannian manifolds. Using the variational definition of Riemannian polynomials, we formulate a high order variational problem on a manifold equipped with a Riemannian metric, which depends on a smoothing parameter and gives rise to what we call smoothing geometric spli...

Fitting smooth paths on riemannian manifolds

In this paper we formulate a least squares problem on a Riemannian manifold M, in order to generate smoothing spline curves fitting a given data set of points in M, q0, q1, . . . , qN, at given instants of time t0 < t1 < • • • < tN. Using tools from Riemannian geometry, we derive the Euler-Lagrange equations associated to this variational problem and prove that its solutions are Riemannian cubic polynomials def...

Optimization on quadratic matrix Lie groups

We study optimality properties of the smooth function tr (Θ-¹QΘN – 2MΘ-¹) , viewed as a function of Θ, with Θ belonging to certain quadratic matrix Lie groups which are generalizations of the orthogonal group. Some optimization matrix problems are formulated in terms of this function. Computational issues based on continuous algorithms are discussed. ; ISR; research network contract FMRXCT-970137

A multi-input/multi-output system representation of generalized splines in R^n

The Moser-Veselov equation

We study the orthogonal solutions of the matrix equation XJ-JXT=M, where J is symmetric positive definite and M is skew-symmetric. This equation arises in the discrete version of the dynamics of a rigid body, investigated by Moser and Veselov [15]. We show connections between orthogonal solutions of this equation and solutions of a certain algebraic Riccati equation. This will bring out the symplectic geometry ...

The De Casteljau Algorithm on Lie Groups and Spheres

We examine the De Casteljau algorithm in the context of Riemannian symmetric spaces. This algorithm, whose classical form is used to generate interpolating polynomials in $$\mathbbR^n $$, was also generalized to arbitrary Riemannian manifolds by others. However, the implementation of the generalized algorithm is difficult since detailed structure, such as boundary value expressions, has not been available. Lie ...

L-splines - A manifestation of optimal control

We show how to generate a class of Euclidean splines, called L-splines, as solutions of a high-order variational problem. We also show connections between L-splines and optimal control theory, leading to the conclusion that L-splines are manifestations of an optimal behavior ; ISR, project ERBFMRXCT970137

On computing real logarithms for matrices in the Lie group of special Euclidean...

We show that the diagonal Pade approximants methods, both for computing the principal logarithm of matrices belonging to the Lie groupSE (n, IR) of special Euclidean motions in IRn and to compute the matrix exponential of elements in the corresponding Lie algebra se(n, IR), are structure preserving. Also, for the particular cases when n == 2,3 we present an alternative closed form to compute the principal logar...

A second order Riemannian variational problem from a Hamiltonian perspective

We present a Hamiltonian formulation of a second order variational problem on a differentiable manifold Q, endowed with a Riemannian metric < .,.> and explore the possibility of writing down the extremal solutions of that problem as a flow in the space TQ T*Q T*Q. For that we utilize the connection r on Q, corresponding to the metric < .,.>. In general the results depend upon a choice of frame for TQ, but for...