146 research outputs found
Iterative Log Thresholding
Sparse reconstruction approaches using the re-weighted l1-penalty have been
shown, both empirically and theoretically, to provide a significant improvement
in recovering sparse signals in comparison to the l1-relaxation. However,
numerical optimization of such penalties involves solving problems with
l1-norms in the objective many times. Using the direct link of reweighted
l1-penalties to the concave log-regularizer for sparsity, we derive a simple
prox-like algorithm for the log-regularized formulation. The proximal splitting
step of the algorithm has a closed form solution, and we call the algorithm
'log-thresholding' in analogy to soft thresholding for the l1-penalty.
We establish convergence results, and demonstrate that log-thresholding
provides more accurate sparse reconstructions compared to both soft and hard
thresholding. Furthermore, the approach can be directly extended to
optimization over matrices with penalty for rank (i.e. the nuclear norm penalty
and its re-weigthed version), where we suggest a singular-value
log-thresholding approach.Comment: 5 pages, 4 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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