Splines of class C<sup>k</sup> on non-euclidean spaces

Camarinha, Margarida; Leite, F. Silva; Crouch, P. E.

doi:10.1093/imamci/12.4.399

Cited by 101 publications

(90 citation statements)

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“…We divide the 25 signals into 5 groups for 5-fold CV. For Full piecewise GGR, we regress on the signals with frequencies in the two intervals (0, 5) and [5,10). The testing MAE ranges from 0.49e-15±0.32e-15 for both GGR (pairwise searching) and Full GGR to 0.58e-15±0.28e-15 for Full piecewise GGR, cf.…”

Section: Resultsmentioning

confidence: 99%

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Geodesic Regression on the Grassmannian

Hong

Kwitt

Singh

et al. 2014

Computer Vision – ECCV 2014

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Abstract. This paper considers the problem of regressing data points on the Grassmann manifold over a scalar-valued variable. The Grassmannian has recently gained considerable attention in the vision community with applications in domain adaptation, face recognition, shape analysis, or the classification of linear dynamical systems. Motivated by the success of these approaches, we introduce a principled formulation for regression tasks on that manifold. We propose an intrinsic geodesic regression model generalizing classical linear least-squares regression. Since geodesics are parametrized by a starting point and a velocity vector, the model enables the synthesis of new observations on the manifold. To exemplify our approach, we demonstrate its applicability on three vision problems where data objects can be represented as points on the Grassmannian: the prediction of traffic speed and crowd counts from dynamical system models of surveillance videos and the modeling of aging trends in human brain structures using an affine-invariant shape representation.

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Section: Resultsmentioning

confidence: 99%

“…While differential geometric concepts, such as geodesics and intrinsic higherorder curves, have been well studied [23,5], their use for regression has only recently gained interest. A variety of methods extending concepts of regression in Euclidean spaces to nonflat manifolds have been proposed.…”

Section: Related Workmentioning

confidence: 99%

Geodesic Regression on the Grassmannian

Hong

Kwitt

Singh

et al. 2014

Computer Vision – ECCV 2014

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“…The equation (1) [32]) and explored from a dynamical interpolation perspective in 1995 (see [17]). Interesting points related to this subject have been developed in the last few years, namely a geometric theory surprisingly close to the Riemannian theory of geodesics (see [2,3,4,12,14,15,16,19,30,31,34,35]). We recall, in particular, a result which says that if V denotes the velocity vector field of a cubic polynomial x, then…”

Section: Introductionmentioning

confidence: 99%

“…In [31], the author solves by quadratures the linking equation on SO(3) and SO(1, 2) of the Riemannian cubics. Finally, [34] studies n-th order generalizations of RCP introduced in [14]. To our knowledge, the first Hamiltonian description of the RCP problem has been considered in [15] (made in collaboration with one of the authors).…”

Section: Introductionmentioning

confidence: 99%

Cubic polynomials on Lie groups: reduction of the Hamiltonian system

Abrunheiro

Camarinha

Clemente-Gallardo

2011

J. Phys. A: Math. Theor.

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Abstract. This paper analyzes the optimal control problem of cubic polynomials on compact Lie groups from a Hamiltonian point of view and its symmetries. The dynamics of the problem is described by a presymplectic formalism associated with the canonical symplectic form on the cotangent bundle of the semidirect product of the Lie group and its Lie algebra. Using these control geometric tools, the relation between the Hamiltonian approach developed here and the known variational one is analyzed. After making explicit the left trivialized system, we use the technique of Marsden-Weinstein reduction to remove the symmetries of the Hamiltonian system. In view of the reduced dynamics, we are able to guarantee, by means of the Lie-Cartan theorem, the existence of a considerable number of independent integrals of motion in involution.

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“…Further generalizations of this work have been developed in a series of papers by Egerstedt and Martin and their collaborators in [1,7,9,11,14] but what is novel in this paper is that we focus explicitly on the convergence aspects of the problem. A somewhat similar approach for interpolating splines has been developed by Crouch and Silva-Leite and their collaborators in [2][3][4][5] with emphasis on deriving splines on non-Euclidean spaces.…”

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confidence: 99%

Statistical Estimates for Generalized Splines

Egerstedt

Martin

2003

ESAIM: COCV

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Abstract. In this paper it is shown that the generalized smoothing spline obtained by solving an optimal control problem for a linear control system converges to a deterministic curve even when the data points are perturbed by random noise. We furthermore show that such a spline acts as a filter for white noise. Examples are constructed that support the practical usefulness of the method as well as gives some hints as to the speed of convergence.Mathematics Subject Classification. 93-xx.

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Splines of class C^k on non-euclidean spaces

Cited by 101 publications

References 0 publications

Geodesic Regression on the Grassmannian

Geodesic Regression on the Grassmannian

Cubic polynomials on Lie groups: reduction of the Hamiltonian system

Statistical Estimates for Generalized Splines

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Splines of class Ck on non-euclidean spaces

Cited by 101 publications

References 0 publications

Geodesic Regression on the Grassmannian

Geodesic Regression on the Grassmannian

Cubic polynomials on Lie groups: reduction of the Hamiltonian system

Statistical Estimates for Generalized Splines

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Product

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Splines of class C^k on non-euclidean spaces