References of "Sepulchre, Rodolphe"
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See detailSensitivity analysis of phase response curves
Sacré, Pierre ULg; Sepulchre, Rodolphe ULg

Conference (2011, May 23)

The Phase Response Curve (PRC) has proven a useful tool for the reduction of complex oscillator models to one-dimensional phase models. We introduce the sensitivity analysis of this important mathematical ... [more ▼]

The Phase Response Curve (PRC) has proven a useful tool for the reduction of complex oscillator models to one-dimensional phase models. We introduce the sensitivity analysis of this important mathematical object and its numerical implementation. As an application, we study simple biochemical models of circadian oscillators and discuss how sensitivity analysis helps drawing connections between the state-space model of the oscillator and its phase response curve. [less ▲]

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See detailSensitivity analysis of phase response curves
Sacré, Pierre ULg; Sepulchre, Rodolphe ULg

Conference (2011, March 16)

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See detailRegression on fixed-rank positive semidefinite matrices: a Riemannian approach
Meyer, Gilles ULg; Bonnabel, Silvère; Sepulchre, Rodolphe ULg

in Journal of Machine Learning Research (2011), 12(Feb), 593625

The paper addresses the problem of learning a regression model parameterized by a fixed-rank positive semidefinite matrix. The focus is on the nonlinear nature of the search space and on scalability to ... [more ▼]

The paper addresses the problem of learning a regression model parameterized by a fixed-rank positive semidefinite matrix. The focus is on the nonlinear nature of the search space and on scalability to high-dimensional problems. The mathematical developments rely on the theory of gradient descent algorithms adapted to the Riemannian geometry that underlies the set of fixed-rank positive semidefinite matrices. In contrast with previous contributions in the literature, no restrictions are imposed on the range space of the learned matrix. The resulting algorithms maintain a linear complexity in the problem size and enjoy important invariance properties. We apply the proposed algorithms to the problem of learning a distance function parameterized by a positive semidefinite matrix. Good performance is observed on classical benchmarks. [less ▲]

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See detailLocal stability results for the collective behaviors of infinite populations of pulse-coupled oscillators
Mauroy, Alexandre ULg; Sepulchre, Rodolphe ULg

in Proceedings of the IEEE Conference on Decision and Control (2011)

In this paper, we investigate the behavior of pulse-coupled integrate-and-fire oscillators. Because the stability analysis of finite populations is intricate, we investigate stability results in the ... [more ▼]

In this paper, we investigate the behavior of pulse-coupled integrate-and-fire oscillators. Because the stability analysis of finite populations is intricate, we investigate stability results in the approximation of infinite populations. In addition to recovering known stability results of finite populations, we also obtain new stability results for infinite populations. In particular, under a weak coupling assumption, we solve for the continuum model a conjecture still prevailing in the finite dimensional case. © 2011 IEEE. [less ▲]

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See detailSynchronization on the circle
Sarlette, Alain ULg; Sepulchre, Rodolphe ULg

in Dubbeldam; Green; Lenstra (Eds.) "The complexity of dynamical systems: a multi-disciplinary perspective" (2011)

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See detailLinear regression under fixed-rank constraints: a Riemannian approach
Meyer, Gilles ULg; Bonnabel, Silvère; Sepulchre, Rodolphe ULg

in Proceedings of the 28th International Conference on Machine Learning (2011)

In this paper, we tackle the problem of learning a linear regression model whose parameter is a fixed-rank matrix. We study the Riemannian manifold geometry of the set of fixed-rank matrices and develop ... [more ▼]

In this paper, we tackle the problem of learning a linear regression model whose parameter is a fixed-rank matrix. We study the Riemannian manifold geometry of the set of fixed-rank matrices and develop efficient line-search algorithms. The proposed algorithms have many applications, scale to high-dimensional problems, enjoy local convergence properties and confer a geometric basis to recent contributions on learning fixed-rank matrices. Numerical experiments on benchmarks suggest that the proposed algorithms compete with the state-of-the-art, and that manifold optimization offers a versatile framework for the design of rank-constrained machine learning algorithms. [less ▲]

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See detailRank-constrained linear regression: a Riemannian approach
Meyer, Gilles ULg; Bonnabel, Silvère; Sepulchre, Rodolphe ULg

Poster (2010, December)

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See detailConsensus in non-commutative spaces
Sepulchre, Rodolphe ULg; Sarlette, Alain ULg; Rouchon, Pierre

in Proceedings of the 49th IEEE Conference on Decision and Control (2010, December)

Convergence analysis of consensus algorithms is revisited in the light of the Hilbert distance. The Lyapunov function used in the early analysis by Tsitsiklis is shown to be the Hilbert distance to ... [more ▼]

Convergence analysis of consensus algorithms is revisited in the light of the Hilbert distance. The Lyapunov function used in the early analysis by Tsitsiklis is shown to be the Hilbert distance to consensus in log coordinates. Birkhoff theorem, which proves contraction of the Hilbert metric for any positive homogeneous monotone map, provides an early yet general convergence result for consensus algorithms. Because Birkhoff theorem holds in arbitrary cones, we extend consensus algorithms to the cone of positive definite matrices. The proposed generalization finds applications in the convergence analysis of quantum stochastic maps, which are a generalization of stochastic maps to non-commutative probability spaces. [less ▲]

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See detailDelayed decision-making in bistable models
Trotta, Laura ULg; Sepulchre, Rodolphe ULg; Bullinger, Eric ULg

in Proceedings of the 49th IEEE Conference on Decision and Control (2010, December)

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See detailConsensus on Nonlinear Spaces
Sepulchre, Rodolphe ULg

in Proceedings of the 8th IFAC Symposium on Nonlinear Control Systems (2010, September)

Consensus problems have attracted significant attention in the control community over the last decade. They act as a rich source of new mathematical problems pertaining to the growing field of cooperative ... [more ▼]

Consensus problems have attracted significant attention in the control community over the last decade. They act as a rich source of new mathematical problems pertaining to the growing field of cooperative and distributed control. This paper is an introduction to consensus problems whose underlying state-space is not a linear space, but instead a highly symmetric nonlinear space such as the circle and other relevant generalizations. A geometric approach is shown to highlight the connection between several fundamental models of consensus, synchronization, and coordination, to raise significant global convergence issues not present in linear models, and to be relevant for a number of engineering applications, including the design of planar or spatial coordinated motions. [less ▲]

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See detailAdaptive filtering for estimation of a low-rank positive semidefinite matrix
Bonnabel, Silvère; Meyer, Gilles ULg; Sepulchre, Rodolphe ULg

in Proceedings of the 19th International Symposium on Mathematical Theory of Networks and Systems (2010, July)

In this paper, we adopt a geometric viewpoint to tackle the problem of estimating a linear model whose parameter is a fixed-rank positive semidefinite matrix. We consider two gradient descent flows ... [more ▼]

In this paper, we adopt a geometric viewpoint to tackle the problem of estimating a linear model whose parameter is a fixed-rank positive semidefinite matrix. We consider two gradient descent flows associated to two distinct Riemannian quotient geometries that underlie this set of matri- ces. The resulting algorithms are non-linear and can be viewed as a generalization of Least Mean Squares that instrically constrain the parameter within the manifold search space. Such algorithms designed for low-rank matrices find applications in high-dimensional distance learning problems for classification or clustering. [less ▲]

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