The Geometry of Low-Rank Kalman Filters - Matrix Information Geometry - page 63

topological obstructions, we proved “eventual contraction” (i.e. contraction after
some finite time for almost any initial condition) in the special case where A is
symmetric. Without this assumption, Oja flow, and thus the low-rank Riccati flow,
do not necessarily converge.

• The filter proposed in this paper requires a reduced number of numerical operations

and storage capacity. It would be of interest to test its efficiency in a particular
large-scale application and to evaluate its relative merits with respect to alternative
approaches that focus on sparsity of the covariance matrix (e.g. [ 22 ]).

Acknowledgments This paper presents research partially supported by the Belgian Programme
on Inter-university Poles of Attraction, initiated by the Belgian State, Prime Minister's Office for
Science, Technology and Culture. The research was partially completed while the second author
was visiting Mines Paris-Tech as an invited professor.

References

1. Absil, P.A., Mahony, R.: Optimization Algorithms on Matrix Manifolds. Princeton University
Press, Princeton (2007)
2. Aghannan, N., Rouchon, P.: An intrinsic observer for a class of lagrangian systems. IEEE
Trans. Autom. Control 48 (6), 936-945 (2003)
3. Bonnabel, S.: A simple intrinsic reduced-observer for geodesic flow. IEEE Trans. Autom.
Control 55 (9), 2186-2191 (2010)
4. Bonnabel, S., Astolfi, A., Sepulchre, R.: Contraction and observer design on cones. In: IEEE
Conference on Decision and, Control 2011 (CDC11), 2011
5. Bonnabel, S., Sepulchre, R.: Riemannian metric and geometric mean for positive semidefinite
matrices of fixed rank. SIAM J. Matrix Anal. Appl. 31 , 1055 (2009)
6. Bougerol, P.: Kalman filtering with random coefficients and contractions. Siam J. Control
Optim. 31 , 942-959 (1993)
7. Boutayeb, M., Rafaralahy, H., Darouach, M.: Convergence analysis of the extended Kalman
filter used as an observer for nonlinear deterministic discrete-time systems. IEEE Trans. Autom.
control 42 , 581-586 (1997)
8. Dee, D.P.: Simplification of the Kalman filter for meteorological data assimilation. Quart. J.
Roy. Meteor. Soc. 10 , 365-384 (1990)
9. Dissanayake, G., Newman, P., Durrant-Whyte, H.F., Clark, S., Csobra, M.: A solution to the
simultaneous localisation and mapping (slam) problem. IEEE Trans. Robot. Autom. 17 , 229-
241 (2001)
10. El Gharamti, M., Hoteit, I., Sun, S.: Low-rank Kalman filtering for efficient state estimation of
subsurface advective contaminant transport models. J. Environ. Eng. 1 , 383 (2011)
11. Faraut, J., Koranyi, A.: Analysis on Symmetric Cones. Oxford University Press, London (1994)
12. Jouffroy, J., Slotine, J.J.E: Methodological remarks on contraction theory. In: 43rd IEEE Con-
ference on Decision and, Control, 2004
13. Journee, M., Absil, P.-A., Bach, F., Sepulchre, R.: Low-rank optimization on the cone of positive
semidefinite matrices. SIAM J. Optim. 20 (5), 2327-2351 (2010)
14. Kalman, R., Bucy, R.: New results in linear filtering and prediction theory. Basic Eng. Trans.
ASME Ser. D 83 (3), 95-108 (1961)
15. Lohmiler, W., Slotine, J.J.E.: On metric analysis and observers for nonlinear systems. Auto-
matica 34 (6), 683-696 (1998)
16. Meyer, G., Bonnabel, S., Sepulchre, R.: Regression on fixed-rank positive semidefinite matri-
ces: a Riemannian approach. J. Mach. Learn. Res. (JMLR) 12 , 593-625 (2011)