R3MC: A Riemannian three-factor algorithm for low-rank matrix completion

We exploit the versatile framework of Riemannian optimization on quotient manifolds to develop R3MC, a nonlinear conjugate-gradient method for low-rank matrix completion. The underlying search space of fixed-rank matrices is endowed with a novel Riemannian metric that is tailored to the least-squares cost. Numerical comparisons suggest that R3MC robustly outperforms state-of-the-art algorithms across different problem instances, especially those that combine scarcely sampled and ill-conditioned data.Comment: Accepted for publication in the proceedings of the 53rd IEEE Conference on Decision and Control, 201

Bursting through interconnection of excitable circuits

We outline the methodology for designing a bursting circuit with robustness and control properties reminiscent of those encountered in biological bursting neurons. We propose that this design question is tractable when addressed through the interconnection theory of two excitable circuits, realized solely with first-order filters and sigmoidal I-V elements. The circuit can be designed and controlled by shaping its I-V curves in the relevant timescales, giving a novel and intuitive methodology for implementing single neuron behaviors in hardware.The research leading to these results has received funding from the European Research Council under the Advanced ERC Grant Agreement Switchlet n.670645

Riemannian preconditioning

This paper exploits a basic connection between sequential quadratic programming and Riemannian gradient optimization to address the general question of selecting a metric in Riemannian optimization, in particular when the Riemannian structure is sought on a quotient manifold. The proposed method is shown to be particularly insightful and efficient in quadratic optimization with orthogonality and/or rank constraints, which covers most current applications of Riemannian optimization in matrix manifolds.Belgium Science Policy Office, FNRS (Belgium)This is the author accepted manuscript. The final version is available from The Society for Industrial and Applied Mathematics via http://dx.doi.org/10.1137/14097086

Differentially Positive Systems

The paper introduces and studies differentially positive systems, that is, systems whose linearization along an arbitrary trajectory is positive. A generalization of Perron Frobenius theory is developed in this differential framework to show that the property induces a (conal) order that strongly constrains the asymptotic behavior of solutions. The results illustrate that behaviors constrained by local order properties extend beyond the well-studied class of linear positive systems and monotone systems, which both require a constant cone field and a linear state space.The research was supported by the Fund for Scientific Research FNRS and by the Engineering and Physical Sciences Research Council under Grant EP/G066477/1.This is the author accepted manuscript. The final version is available from IEEE via http://dx.doi.org/10.1109/TAC.2015.243752

Positivity, monotonicity, and consensus on lie groups

Dynamical systems whose linearizations along trajectories are positive in the sense that they infinitesimally contract a smooth cone field are called differentially positive. The property can be thought of as a generalization of monotonicity, which is differential positivity in a linear space with respect to a constant cone field. Differential positivity places significant constraints on the asymptotic behavior of trajectories under mild technical conditions. This paper studies differentially positive systems defined on Lie groups. The geometry of a Lie group allows for the generation of invariant cone fields over the tangent bundle given a single cone in the Lie algebra. We outline the mathematical framework for studying differential positivity of discrete and continuous-time dynamics on a Lie group with respect to an invariant cone field and motivate the use of this analysis framework in nonlinear control, and, in particular in nonlinear consensus theory. We also introduce a generalized notion of differential positivity of a dynamical system with respect to an extended notion of cone fields generated by cones of rank k. This new property provides the basis for a generalization of differential Perron-Frobenius theory, whereby the Perron-Frobenius vector field which shapes the one-dimensional attractors of a differentially positive system is replaced by a distribution of rank k that results in k-dimensional integral submanifold attractors instead

Ordering positive definite matrices

We introduce new partial orders on the set $S^+_n$ of positive definite matrices of dimension $n$ derived from the affine-invariant geometry of $S^+_n$. The orders are induced by affine-invariant cone fields, which arise naturally from a local analysis of the orders that are compatible with the homogeneous geometry of $S^+_n$ defined by the natural transitive action of the general linear group $GL(n)$. We then take a geometric approach to the study of monotone functions on $S^+_n$ and establish a number of relevant results, including an extension of the well-known L\"owner-Heinz theorem derived using differential positivity with respect to affine-invariant cone fields