References of "Sepulchre, Rodolphe"
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See detailAn Organizing Center in a Planar Model of Neuronal Excitability
Franci, Alessio ULg; Drion, Guillaume ULg; Sepulchre, Rodolphe ULg

in SIAM Journal on Applied Dynamical Systems (2012), 11(4), 1698-1722

The paper studies the excitability properties of a generalized FitzHugh-Nagumo model. The model differs from the purely competitive FitzHugh-Nagumo model in that it accounts for the effect of cooperative ... [more ▼]

The paper studies the excitability properties of a generalized FitzHugh-Nagumo model. The model differs from the purely competitive FitzHugh-Nagumo model in that it accounts for the effect of cooperative gating variables such as activation of calcium currents. Excitability is explored by unfolding a pitchfork bifurcation that is shown to organize five different types of excitability. In addition to the three classical types of neuronal excitability, two novel types are described and distinctly associated to the presence of cooperative variables. [less ▲]

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See detailFixed-rank matrix factorizations and Riemannian low-rank optimization
Mishra, Bamdev ULg; Meyer, Gilles ULg; Bonnabel, Silvere et al

E-print/Working paper (2012)

Motivated by the problem of learning a linear regression model whose parameter is a large fixed-rank non-symmetric matrix, we consider the optimization of a smooth cost function defined on the set of ... [more ▼]

Motivated by the problem of learning a linear regression model whose parameter is a large fixed-rank non-symmetric matrix, we consider the optimization of a smooth cost function defined on the set of fixed-rank matrices. We adopt the geometric optimization framework of optimization on Riemannian matrix manifolds. We study the underlying geometries of several well-known fixed-rank matrix factorizations and then exploit the Riemannian geometry of the search space in the design of a class of gradient descent and trust-region algorithms. The proposed algorithms generalize our previous results on fixed-rank symmetric positive semidefinite matrices, apply to a broad range of applications, scale to high-dimensional problems and confer a geometric basis to recent contributions on the learning of fixed-rank non-symmetric matrices. We make connections with existing algorithms in the context of low-rank matrix completion and discuss relative usefulness of the proposed framework. Numerical experiments suggest that the proposed algorithms compete with the state-of-the-art and that manifold optimization offers an effective and versatile framework for the design of machine learning algorithms that learn a fixed-rank matrix. [less ▲]

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See detailA Riemannian geometry for low-rank matrix completion
Mishra, Bamdev ULg; Karavadi, Adithya Apuroop; Sepulchre, Rodolphe ULg

E-print/Working paper (2012)

We propose a new Riemannian geometry for fixed-rank matrices that is specifically tailored to the low-rank matrix completion problem. Exploiting the degree of freedom of a quotient space, we tune the ... [more ▼]

We propose a new Riemannian geometry for fixed-rank matrices that is specifically tailored to the low-rank matrix completion problem. Exploiting the degree of freedom of a quotient space, we tune the metric on our search space to the particular least square cost function. At one level, it illustrates in a novel way how to exploit the versatile framework of optimization on quotient manifold. At another level, our algorithm can be considered as an improved version of LMaFit, the state-of-the-art Gauss-Seidel algorithm. We develop necessary tools needed to perform both first-order and second-order optimization. In particular, we propose gradient descent schemes (steepest descent and conjugate gradient) and trust-region algorithms. We also show that, thanks to the simplicity of the cost function, it is numerically cheap to perform an exact linesearch given a search direction, which makes our algorithms competitive with the state-of-the-art on standard low-rank matrix completion instances. [less ▲]

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See detailLow-rank optimization with trace norm penalty
Mishra, Bamdev ULg; Meyer, Gilles ULg; Bach, Francis et al

E-print/Working paper (2012)

The paper addresses the problem of low-rank trace norm minimization. We propose an algorithm that alternates between fixed-rank optimization and rank-one updates. The fixed-rank optimization is ... [more ▼]

The paper addresses the problem of low-rank trace norm minimization. We propose an algorithm that alternates between fixed-rank optimization and rank-one updates. The fixed-rank optimization is characterized by an efficient factorization that makes the trace norm differentiable in the search space and the computation of duality gap numerically tractable. The search space is nonlinear but is equipped with a particular Riemannian structure that leads to efficient computations. We present a second-order trust-region algorithm with a guaranteed quadratic rate of convergence. Overall, the proposed optimization scheme converges super-linearly to the global solution while still maintaining complexity that is linear in the number of rows of the matrix. To compute a set of solutions efficiently for a grid of regularization parameters we propose a predictor-corrector approach on the quotient manifold that outperforms the naive warm-restart approach. The performance of the proposed algorithm is illustrated on problems of low-rank matrix completion and multivariate linear regression. [less ▲]

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See detailMatching an oscillator model to a phase response curve
Sacré, Pierre ULg; Sepulchre, Rodolphe ULg

in Proceedings of the Joint 50th IEEE Conference on Decision and Control and European Control Conference (2011, December)

The Phase Response Curve (PRC) has proven a useful tool for the reduction of complex oscillator models. It is also an information often experimentally available to the biologist. This paper introduces a ... [more ▼]

The Phase Response Curve (PRC) has proven a useful tool for the reduction of complex oscillator models. It is also an information often experimentally available to the biologist. This paper introduces a numerical tool based on the sensitivity analysis of the PRC to adapt initial model parameters in order to match a particular PRC shape. We illustrate the approach on a simple biochemical model of circadian oscillator. [less ▲]

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See detailLow-rank optimization for distance matrix completion
Mishra, Bamdev ULg; Meyer, Gilles ULg; Sepulchre, Rodolphe ULg

in Proceedings of the 50th IEEE Conference on Decision and Control (2011, December)

This paper addresses the problem of low-rank distance matrix completion. This problem amounts to recover the missing entries of a distance matrix when the dimension of the data embedding space is possibly ... [more ▼]

This paper addresses the problem of low-rank distance matrix completion. This problem amounts to recover the missing entries of a distance matrix when the dimension of the data embedding space is possibly unknown but small compared to the number of considered data points. The focus is on high-dimensional problems. We recast the considered problem into an optimization problem over the set of low-rank positive semidefinite matrices and propose two efficient algorithms for low-rank distance matrix completion. In addition, we propose a strategy to determine the dimension of the embedding space. The resulting algorithms scale to high-dimensional problems and monotonically converge to a global solution of the problem. Finally, numerical experiments illustrate the good performance of the proposed algorithms on benchmarks. [less ▲]

Detailed reference viewed: 49 (8 ULg)
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 ▲]

Detailed reference viewed: 90 (13 ULg)