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How modeling can reconcile apparently discrepant experimental results: the case of pacemaking in dopaminergic neurons. Drion, Guillaume ; Massotte, Laurent ; Sepulchre, Rodolphe et al in PLoS Computational Biology (2011), 7(5), 1002050 Detailed reference viewed: 129 (40 ULg)Linear regression under fixed-rank constraints: a Riemannian approach Meyer, Gilles ; ; Sepulchre, Rodolphe 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 ▲] Detailed reference viewed: 152 (4 ULg)Rank-constrained linear regression: a Riemannian approach Meyer, Gilles ; ; Sepulchre, Rodolphe Poster (2010, December) Detailed reference viewed: 18 (0 ULg)Consensus in non-commutative spaces Sepulchre, Rodolphe ; Sarlette, Alain ; 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 ▲] Detailed reference viewed: 64 (5 ULg)Delayed decision-making in bistable models Trotta, Laura ; Sepulchre, Rodolphe ; Bullinger, Eric in Proceedings of the 49th IEEE Conference on Decision and Control (2010, December) Detailed reference viewed: 141 (66 ULg)SK channels as regulators of synaptically-induced bursting and neural synchrony Drion, Guillaume ; Collard, Anne ; Seutin, Vincent et al Poster (2010, October) Detailed reference viewed: 9 (0 ULg)Consensus on Nonlinear Spaces Sepulchre, Rodolphe 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 ▲] Detailed reference viewed: 93 (14 ULg)Adaptive filtering for estimation of a low-rank positive semidefinite matrix ; Meyer, Gilles ; Sepulchre, Rodolphe 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 ▲] Detailed reference viewed: 52 (16 ULg)Global Analysis of Firing Maps Mauroy, 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 ▲] Detailed reference viewed: 74 (18 ULg)Multiple feedback loops in circadian cycles: robustness and entrainment as selection criteria ; Sacré, Pierre ; 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 ▲] Detailed reference viewed: 127 (39 ULg)Géométrie des matrices positives semi-définies de rang fixé : un peu de théorie et beaucoup d’applications Sepulchre, Rodolphe ; ; 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 ▲] Detailed reference viewed: 75 (7 ULg)Low-rank optimization on the cone of positive semidefinite matrices Journee, Michel ; ; et al in SIAM Journal on Optimization (2010), 20(5) Detailed reference viewed: 162 (12 ULg)Coordinated motion design on Lie groups Sarlette, Alain ; ; Sepulchre, Rodolphe 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 ▲] Detailed reference viewed: 67 (19 ULg)Selection of circadian clock models for robust entrainment: an analysis based on the phase response curve Sacré, Pierre ; ; et al Conference (2010, March 30) Detailed reference viewed: 59 (30 ULg)SK Channels as Regulators of Synaptically Induced Bursting and Neural Synchrony Drion, Guillaume ; Collard, Anne ; Seutin, Vincent et al Conference (2010, March) Detailed reference viewed: 10 (0 ULg)Means and medians in nonlinear spaces Collard, Anne ; Sepulchre, Rodolphe Conference (2010, March) Detailed reference viewed: 15 (5 ULg)Generalized power method for sparse principal component analysis ; ; Journee, Michel et al in Journal of Machine Learning Research (2010), 11 Detailed reference viewed: 122 (34 ULg)Modeling and Measurements of Cerebral Signaling Circuits Drion, Guillaume ; Seutin, Vincent ; Sepulchre, Rodolphe Conference (2010) Detailed reference viewed: 2 (0 ULg)Global analysis of pulse-coupled oscillators: discrete and continuous models Mauroy, Alexandre ; Sepulchre, Rodolphe Conference (2010) Detailed reference viewed: 2 (0 ULg)M-type channels selectively control bursting in rat dopaminergic neurons Drion, Guillaume ; ; Waroux, Olivier et al in European Journal of Neuroscience (2010), 31 Detailed reference viewed: 127 (60 ULg) |
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