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Low-rank optimization with trace norm penalty Mishra, Bamdev ; Meyer, Gilles ; 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 ▲] Detailed reference viewed: 58 (23 ULg)Contraction of monotone phase-coupled oscillators Mauroy, Alexandre ; Sepulchre, Rodolphe in Systems & Control Letters (2012), 61(11), 1097-1102 This paper establishes a global contraction property for networks of phase-coupled oscillators characterized by a monotone coupling function. The contraction measure is a total variation distance. The ... [more ▼] This paper establishes a global contraction property for networks of phase-coupled oscillators characterized by a monotone coupling function. The contraction measure is a total variation distance. The contraction property determines the asymptotic behavior of the network, which is either finite-time synchronization or asymptotic convergence to a splay state. © 2012 Elsevier B.V. All rights reserved. [less ▲] Detailed reference viewed: 10 (0 ULg)Fixed-rank matrix factorizations and Riemannian low-rank optimization Mishra, Bamdev ; Meyer, Gilles ; 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 ▲] Detailed reference viewed: 41 (5 ULg)A Riemannian geometry for low-rank matrix completion Mishra, Bamdev ; ; Sepulchre, Rodolphe 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 ▲] Detailed reference viewed: 90 (18 ULg)Matching an oscillator model to a phase response curve Sacré, Pierre ; Sepulchre, Rodolphe 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 ▲] Detailed reference viewed: 72 (19 ULg)Low-rank optimization for distance matrix completion Mishra, Bamdev ; Meyer, Gilles ; Sepulchre, Rodolphe 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: 57 (11 ULg)How Modeling can Reconcile Apparently Discrepant Experimental Results: the Case of Pacemaking in Dopaminergic Neurons Drion, Guillaume ; Massotte, Laurent ; Scuvée-Moreau, Jacqueline et al Poster (2011, November) Detailed reference viewed: 12 (0 ULg)Biological bistable models a local analysis for non local predictions Trotta, Laura ; Sepulchre, Rodolphe ; Bullinger, Eric Conference (2011, June 17) Detailed reference viewed: 36 (10 ULg)Sensitivity analysis of phase response curves Sacré, Pierre ; Sepulchre, Rodolphe 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 ▲] Detailed reference viewed: 83 (24 ULg)Delays in the apoptotic switch: look for the hidden saddle Trotta, Laura ; Sepulchre, Rodolphe ; Bullinger, Eric Poster (2011, March 21) Detailed reference viewed: 24 (2 ULg)On the Origin of the Pacemaker Activity of Midbrain Dopaminergic Neurons Drion, Guillaume ; Sepulchre, Rodolphe ; Seutin, Vincent Conference (2011, March 18) Detailed reference viewed: 12 (0 ULg)Sensitivity analysis of phase response curves Sacré, Pierre ; Sepulchre, Rodolphe Conference (2011, March 16) Detailed reference viewed: 38 (17 ULg)Regression on fixed-rank positive semidefinite matrices: a Riemannian approach Meyer, Gilles ; ; Sepulchre, Rodolphe 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 ▲] Detailed reference viewed: 109 (18 ULg)How Modeling can Reconcile Apparently Discrepant Experimental Results: the Case of Pacemaking in Dopaminergic Neurons Drion, Guillaume ; Massotte, Laurent ; Sepulchre, Rodolphe et al Conference (2011, March) Detailed reference viewed: 13 (0 ULg)Selecting an appropriate metric for the processing of Diffusion Tensor Images Collard, Anne ; Sepulchre, Rodolphe Conference (2011, March) Detailed reference viewed: 25 (5 ULg)Existence and stability of a dichotomic behavior in infinite populations of pulse-coupled oscillators Mauroy, Alexandre ; Sepulchre, Rodolphe Poster (2011) Detailed reference viewed: 4 (0 ULg)A transport equation for pulse-coupled phase oscillators and a Lyapunov function for its global analysis Mauroy, Alexandre ; Sepulchre, Rodolphe Conference (2011) Detailed reference viewed: 2 (0 ULg)Existence and stability of a dichotomic behavior in infinite populations of pulse-coupled oscillators Mauroy, Alexandre ; Sepulchre, Rodolphe Conference (2011) Detailed reference viewed: 3 (0 ULg)Local stability results for the collective behaviors of infinite populations of pulse-coupled oscillators Mauroy, Alexandre ; Sepulchre, Rodolphe 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 ▲] Detailed reference viewed: 6 (0 ULg)Synchronization on the circle Sarlette, Alain ; Sepulchre, Rodolphe in Dubbeldam; Green; Lenstra (Eds.) "The complexity of dynamical systems: a multi-disciplinary perspective" (2011) Detailed reference viewed: 71 (3 ULg) |
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