References of "Sepulchre, Rodolphe"
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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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See detailGlobal Analysis of Firing Maps
Mauroy, Alexandre ULg; Hendrickx, Julien; Megretski, Alexandre et al

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

In this paper, we study the behavior of pulse-coupled integrate-and-fire oscillators. Each oscillator is characterized by a state evolving between two threshold values. As the state reaches the upper ... [more ▼]

In this paper, we study the behavior of pulse-coupled integrate-and-fire oscillators. Each oscillator is characterized by a state evolving between two threshold values. As the state reaches the upper threshold, it is reset to the lower threshold and emits a pulse which increments by a constant value the state of every other oscillator. The behavior of the system is described by the so-called firing map: depending on the stability of the firing map, an important dichotomy characterizes the behavior of the oscillators (synchronization or clustering). The firing map is the composition of a linear map with a scalar nonlinearity. After briefly discussing the case of the scalar firing map (corresponding to two oscillators), the stability analysis is extended to the general n-dimensional firing map (for n +1 oscillators). Different models are considered (leaky oscillators, quadratic oscillators,...), with a particular emphasis on the persistence of the dichotomy in higher dimensions. [less ▲]

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See detailMultiple feedback loops in circadian cycles: robustness and entrainment as selection criteria
Hafner, Marc; Sacré, Pierre ULg; Symul, Laura et al

in Proceedings of the Seventh International Workshop on Computational Systems Biology, WCSB 2010 (2010, June)

This paper discusses the contribution of an additional feedback loop to the entrainment and the robustness of a circadian system. To quantify robustness, we perform a global analysis of the system's ... [more ▼]

This paper discusses the contribution of an additional feedback loop to the entrainment and the robustness of a circadian system. To quantify robustness, we perform a global analysis of the system's parameter space. We quantify the parameter region where the circuit displays an experimentally observed behavior, under entrainment. This global measure is comleted with a classification based on the phase response curve (PRC). For two models of circadian rhythms, we found that the one with two loops is more robust than the one with a single loop: the two-loop model shows better resilience to parameter perturbations and it has also a larger region where the PRC matches experimental PRCs of circadian oscillators. [less ▲]

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See detailGéométrie des matrices positives semi-définies de rang fixé : un peu de théorie et beaucoup d’applications
Sepulchre, Rodolphe ULg; Absil, Pierre-Antoine; Bonnabel, Silvère

in Proceedings of Sixième Conférence Internationale Francophone d'Automatique (CIFA 2010) (2010, June)

Cet article est une introduction au calcul et `a l’optimisation sur les matrices sym´etriques positives semid ´efinies de rang (faible) fix´e. L’approche propos´ee est bas´ee sur deux g´eom´etries ... [more ▼]

Cet article est une introduction au calcul et `a l’optimisation sur les matrices sym´etriques positives semid ´efinies de rang (faible) fix´e. L’approche propos´ee est bas´ee sur deux g´eom´etries riemanniennes quotient, qui permettent de calculer efficacement tout en pr´eservant le rang et le caract` ere positif des matrices consid´er´ees. Le champ d’applications est vaste, et l’article survole quelques d´eveloppements r´ecents qui illustrent l’int´erˆet de l’approche consid´er´ee dans les probl`emes de tr`es grande taille rencontr´es en contrˆole, statistiques, et apprentissage. [less ▲]

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See detailLow-rank optimization on the cone of positive semidefinite matrices
Journee, Michel ULg; Bach, F.; Absil, P.-A. et al

in SIAM Journal on Optimization (2010), 20(5)

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See detailCoordinated motion design on Lie groups
Sarlette, Alain ULg; Bonnabel, Silvere; Sepulchre, Rodolphe ULg

in IEEE Transactions on Automatic Control (2010), 55(5), 1047-1058

The present paper proposes a unified geometric framework for coordinated motion on Lie groups. It first gives a general problem formulation and analyzes ensuing conditions for coordinated motion. Then, it ... [more ▼]

The present paper proposes a unified geometric framework for coordinated motion on Lie groups. It first gives a general problem formulation and analyzes ensuing conditions for coordinated motion. Then, it introduces a precise method to design control laws in fully actuated and underactuated settings with simple integrator dynamics. It thereby shows that coordination can be studied in a systematic way once the Lie group geometry of the configuration space is well characterized. Applying the proposed general methodology to particular examples allows to retrieve control laws that have been proposed in the literature on intuitive grounds. A link with Brockett's double bracket flows is also made. The concepts are illustrated on SO(3) , SE(2) and SE(3). [less ▲]

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See detailSelection of circadian clock models for robust entrainment: an analysis based on the phase response curve
Sacré, Pierre ULg; Hafner, Marc; Heinz, Koeppl et al

Conference (2010, March 30)

Detailed reference viewed: 55 (28 ULg)
See detailMeans and medians in nonlinear spaces
Collard, Anne ULg; Sepulchre, Rodolphe ULg

Conference (2010, March)

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See detailGeneralized power method for sparse principal component analysis
Nesterov, Yurii; Richtarik, Peter; Journee, Michel ULg et al

in Journal of Machine Learning Research (2010), 11

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See detailM-type channels selectively control bursting in rat dopaminergic neurons
Drion, Guillaume ULg; Bonjean, Maxime; Waroux, Olivier ULg et al

in European Journal of Neuroscience (2010), 31

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See detailOptimal Data Fitting on Lie Groups: a Coset Approach
Lageman, Christian; Sepulchre, Rodolphe ULg

in Diehl, M. (Ed.) Recent Advances in Optimization and its Applications in Engineering (2010)

This work considers the problem of fitting data on a Lie group by a coset of a compact subgroup. This problem can be seen as an extension of the problem of fitting affine subspaces in Rn to data which can ... [more ▼]

This work considers the problem of fitting data on a Lie group by a coset of a compact subgroup. This problem can be seen as an extension of the problem of fitting affine subspaces in Rn to data which can be solved using principal component analysis. We show how the fitting problem can be reduced for biinvariant distances to a generalized mean calculation on an homogeneous space. For biinvariant Riemannian distances we provide an algorithm based on the Karcher mean gradient algorithm. We illustrate our approach by some examples on SO(n). [less ▲]

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See detailRefining Sparse Principal Components
Journée, M.; Bach, Francis; Absil, P.-A. et al

in Diehl, M. (Ed.) Recent Advances in Optimization and its Applications in Engineering (2010)

In this paper, we discuss methods to refine locally optimal solutions of sparse PCA. Starting from a local solution obtained by existing algorithms, these methods take advantage of convex relaxations of ... [more ▼]

In this paper, we discuss methods to refine locally optimal solutions of sparse PCA. Starting from a local solution obtained by existing algorithms, these methods take advantage of convex relaxations of the sparse PCA problem to propose a refined solution that is still locally optimal but with a higher objective value. [less ▲]

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See detailOptimization on manifolds : methods and applications
Absil, P.-A.; Mahony, R.; Sepulchre, Rodolphe ULg

in Recent Advances in Optimization and its Applications in Engineering (2010)

Summary. This paper provides an introduction to the topic of optimization on manifolds. The approach taken uses the language of differential geometry, however, we choose to emphasise the intuition of the ... [more ▼]

Summary. This paper provides an introduction to the topic of optimization on manifolds. The approach taken uses the language of differential geometry, however, we choose to emphasise the intuition of the concepts and the structures that are important in generating practical numerical algorithms rather than the technical details of the formulation. There are a number of algorithms that can be applied to solve such problems and we discuss the steepest descent and Newton’s method in some detail as well as referencing the more important of the other approaches. There are a wide range of potential applications that we are aware of, and we briefly discuss these applications, as well as explaining one or two in more detail. [less ▲]

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See detailControlling the phase of an oscillator: a phase response curve approach
Efimov, Denis; Sacré, Pierre ULg; Sepulchre, Rodolphe ULg

in Proceedings of the Joint 48th IEEE Conference on Decision and Control and 28th Chinese Control Conference (2009, December)

The paper discusses elementary control strategies to control the phase of an oscillator. Both feedforward and feedback (P and PI) control laws are designed based on the phase response curve (PRC ... [more ▼]

The paper discusses elementary control strategies to control the phase of an oscillator. Both feedforward and feedback (P and PI) control laws are designed based on the phase response curve (PRC) calculated from the linearized model. The performance is evaluated on a popular model of circadian oscillations. [less ▲]

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See detailSynchronization with partial state feedback on SO(n),
Lageman, Christian; Sarlette, Alain ULg; Sepulchre, Rodolphe ULg

in Proceedings of the 48th IEEE Conference on Decision and Control (2009, December)

This paper considers the problem of constructing a distributed feedback law to achieve synchronization for a group of k agents whose states evolve on SO(n) and which exchange only partial state ... [more ▼]

This paper considers the problem of constructing a distributed feedback law to achieve synchronization for a group of k agents whose states evolve on SO(n) and which exchange only partial state information along communication links. The partial state information is given by the action of the state on reference vectors in Rn. We propose a gradient based control law which achieves exponential local convergence to a synchronization configuration under a rank condition on a generalized Laplacian matrix. Furthermore, we discuss the case of time-varying reference vectors and provide a convergence result for this case. The latter helps reach synchronization, requiring less communication links and weaker conditions on the instantaneous reference vectors. Our methods are illustrated on an attitude synchronization problem where agents exchange only their relative positions observed in the respective body frames. [less ▲]

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See detailConsensus on homogeneous manifolds
Sarlette, Alain ULg; Sepulchre, Rodolphe ULg

in Proceedings of the 48th IEEE Conference on Decision and Control (2009, December)

The present paper considers distributed consensus algorithms for agents evolving on a connected compact homogeneous (CCH) manifold. The agents track no external reference and communicate their relative ... [more ▼]

The present paper considers distributed consensus algorithms for agents evolving on a connected compact homogeneous (CCH) manifold. The agents track no external reference and communicate their relative state according to an interconnection graph. The paper first formalizes the consensus problem for synchronization (i.e. maximizing the consensus) and balancing (i.e. minimizing the consensus); it thereby introduces the induced arithmetic mean, an easily computable mean position on CCH manifolds. Then it proposes and analyzes various consensus algorithms on manifolds: natural gradient algorithms which reach local consensus equilibria; an adaptation using auxiliary variables for almost-global synchronization or balancing; and a stochastic gossip setting for global synchronization. It closes by investigating the dependence of synchronization properties on the attraction function between interacting agents on the circle. The theory is also illustrated on SO(n) and on the Grassmann manifolds. [less ▲]

Detailed reference viewed: 39 (7 ULg)