Learning the Fréchet Mean over the Manifold of Symmetric Positive-Definite Matrices

Fiori, Simone

doi:10.1007/s12559-009-9026-7

Cited by 28 publications

(18 citation statements)

References 19 publications

(27 reference statements)

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“…The differential system (53) is formally equivalent to the one studied in [10], hence the analysis of the improvement due to second-order dynamics over first-order dynamics in approaching a local minimum of the potential energy function leads to the same conclusions as in [10]. Such conclusions may be summarized as follows.…”

Section: Arrange In a Hessian Matrixmentioning

confidence: 88%

Extended Hamiltonian Learning on Riemannian Manifolds: Theoretical Aspects

Fiori

2011

IEEE Trans. Neural Netw.

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This paper introduces a general theory of extended Hamiltonian (second-order) learning on Riemannian manifolds, as an instance of learning by constrained criterion optimization. The dynamical learning equations are derived within the general framework of extended-Hamiltonian stationary-action principle and are expressed in a coordinate-free fashion. A theoretical analysis is carried out in order to compare the features of the dynamical learning theory with the features exhibited by the gradient-based ones. In particular, gradient-based learning is shown to be an instance of dynamical learning, and the classical gradient-based learning modified by a "momentum" term is shown to resemble discrete-time dynamical learning. Moreover, the convergence features of gradient-based and dynamical learning are compared on a theoretical basis. This paper discusses cases of learning by dynamical systems on manifolds of interest in the scientific literature, namely, the Stiefel manifold, the special orthogonal group, the Grassmann manifold, the group of symmetric positive definite matrices, the generalized flag manifold, and the real symplectic group of matrices.

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Section: Arrange In a Hessian Matrixmentioning

confidence: 88%

Extended Hamiltonian Learning on Riemannian Manifolds: Theoretical Aspects

Fiori

2011

IEEE Trans. Neural Netw.

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“…• Representations in terms of elements of the space of the upper triangular matrices with positive diagonal elements, and of the space of the upper triangular matrices with positive diagonal elements whose product is unitary, find applications in the representation of shapes in pattern recognition [17]. It is also worth mentioning the noncompact manifold formed by symmetric positive-definite matrices , which finds a large number of applications [8]. Such a group manifold is one of the few encountered exceptions of noncompact manifolds that admit closed-form expressions of geodesic arcs and geodesic distances when endowed with a Riemannian metric.…”

Section: Further Noncompact Manifolds Of Interestmentioning

confidence: 99%

Learning by Natural Gradient on Noncompact Matrix-Type Pseudo-Riemannian Manifolds

Fiori

2010

IEEE Trans. Neural Netw.

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This paper deals with learning by natural-gradient optimization on noncompact manifolds. In a Riemannian manifold, the calculation of entities such as the closed form of geodesic curves over noncompact manifolds might be infeasible. For this reason, it is interesting to study the problem of learning by optimization over noncompact manifolds endowed with pseudo-Riemannian metrics, which may give rise to tractable calculations. A general theory for natural-gradient-based learning on noncompact manifolds as well as specific cases of interest of learning are discussed.

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“…In [2], the authors concern the statistics of covariance matrices for biomedical engineering applications. The current research mainly focuses on the compact Lie groups, such as the special orthogonal group SO(n) [3] and the unitary group U(n) [4], or other matrix manifolds, such as the special Euclidean group SE(n, R) [5], Grassmann manifold Gr(n, k) [6,7], the Stiefel manifold St(n, k) [8,9] and the symmetry positive-definite matrix manifold SPD(n) [10,11]. However, there is little research on averaging over the Lorentz group O(n, k).…”

Section: Introductionmentioning

confidence: 99%

A Geodesic-Based Riemannian Gradient Approach to Averaging on the Lorentz Group

Wang

Sun

2017

Entropy

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Abstract:In this paper, we propose an efficient algorithm to solve the averaging problem on the Lorentz group O(n, k). Firstly, we introduce the geometric structures of O(n, k) endowed with a Riemannian metric where geodesic could be written in closed form. Then, the algorithm is presented based on the Riemannian-steepest-descent approach. Finally, we compare the above algorithm with the Euclidean gradient algorithm and the extended Hamiltonian algorithm. Numerical experiments show that the geodesic-based Riemannian-steepest-descent algorithm performs the best in terms of the convergence rate.

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Learning the Fréchet Mean over the Manifold of Symmetric Positive-Definite Matrices

Cited by 28 publications

References 19 publications

Extended Hamiltonian Learning on Riemannian Manifolds: Theoretical Aspects

Extended Hamiltonian Learning on Riemannian Manifolds: Theoretical Aspects

Learning by Natural Gradient on Noncompact Matrix-Type Pseudo-Riemannian Manifolds

A Geodesic-Based Riemannian Gradient Approach to Averaging on the Lorentz Group

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