2,809 research outputs found
Convergence analysis of a proximal Gauss-Newton method
An extension of the Gauss-Newton algorithm is proposed to find local
minimizers of penalized nonlinear least squares problems, under generalized
Lipschitz assumptions. Convergence results of local type are obtained, as well
as an estimate of the radius of the convergence ball. Some applications for
solving constrained nonlinear equations are discussed and the numerical
performance of the method is assessed on some significant test problems
Optimization viewpoint on Kalman smoothing, with applications to robust and sparse estimation
In this paper, we present the optimization formulation of the Kalman
filtering and smoothing problems, and use this perspective to develop a variety
of extensions and applications. We first formulate classic Kalman smoothing as
a least squares problem, highlight special structure, and show that the classic
filtering and smoothing algorithms are equivalent to a particular algorithm for
solving this problem. Once this equivalence is established, we present
extensions of Kalman smoothing to systems with nonlinear process and
measurement models, systems with linear and nonlinear inequality constraints,
systems with outliers in the measurements or sudden changes in the state, and
systems where the sparsity of the state sequence must be accounted for. All
extensions preserve the computational efficiency of the classic algorithms, and
most of the extensions are illustrated with numerical examples, which are part
of an open source Kalman smoothing Matlab/Octave package.Comment: 46 pages, 11 figure
Smoothing Dynamic Systems with State-Dependent Covariance Matrices
Kalman filtering and smoothing algorithms are used in many areas, including
tracking and navigation, medical applications, and financial trend filtering.
One of the basic assumptions required to apply the Kalman smoothing framework
is that error covariance matrices are known and given. In this paper, we study
a general class of inference problems where covariance matrices can depend
functionally on unknown parameters. In the Kalman framework, this allows
modeling situations where covariance matrices may depend functionally on the
state sequence being estimated. We present an extended formulation and
generalized Gauss-Newton (GGN) algorithm for inference in this context. When
applied to dynamic systems inference, we show the algorithm can be implemented
to preserve the computational efficiency of the classic Kalman smoother. The
new approach is illustrated with a synthetic numerical example.Comment: 8 pages, 1 figur
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