2,569 research outputs found
Analyzing Least Squares and Kalman Filtered Compressed Sensing
In recent work, we studied the problem of causally reconstructing time
sequences of spatially sparse signals, with unknown and slow time-varying
sparsity patterns, from a limited number of linear "incoherent" measurements.
We proposed a solution called Kalman Filtered Compressed Sensing (KF-CS). The
key idea is to run a reduced order KF only for the current signal's estimated
nonzero coefficients' set, while performing CS on the Kalman filtering error to
estimate new additions, if any, to the set. KF may be replaced by Least Squares
(LS) estimation and we call the resulting algorithm LS-CS. In this work, (a) we
bound the error in performing CS on the LS error and (b) we obtain the
conditions under which the KF-CS (or LS-CS) estimate converges to that of a
genie-aided KF (or LS), i.e. the KF (or LS) which knows the true nonzero sets.Comment: Proc. IEEE Intl. Conf. Acous. Speech Sig. Proc. (ICASSP), 200
Nonlinear Compressive Particle Filtering
Many systems for which compressive sensing is used today are dynamical. The
common approach is to neglect the dynamics and see the problem as a sequence of
independent problems. This approach has two disadvantages. Firstly, the
temporal dependency in the state could be used to improve the accuracy of the
state estimates. Secondly, having an estimate for the state and its support
could be used to reduce the computational load of the subsequent step. In the
linear Gaussian setting, compressive sensing was recently combined with the
Kalman filter to mitigate above disadvantages. In the nonlinear dynamical case,
compressive sensing can not be used and, if the state dimension is high, the
particle filter would perform poorly. In this paper we combine one of the most
novel developments in compressive sensing, nonlinear compressive sensing, with
the particle filter. We show that the marriage of the two is essential and that
neither the particle filter or nonlinear compressive sensing alone gives a
satisfying solution.Comment: Accepted to CDC 201
LS-CS-residual (LS-CS): Compressive Sensing on Least Squares Residual
We consider the problem of recursively and causally reconstructing time
sequences of sparse signals (with unknown and time-varying sparsity patterns)
from a limited number of noisy linear measurements. The sparsity pattern is
assumed to change slowly with time. The idea of our proposed solution,
LS-CS-residual (LS-CS), is to replace compressed sensing (CS) on the
observation by CS on the least squares (LS) residual computed using the
previous estimate of the support. We bound CS-residual error and show that when
the number of available measurements is small, the bound is much smaller than
that on CS error if the sparsity pattern changes slowly enough. We also obtain
conditions for "stability" of LS-CS over time for a signal model that allows
support additions and removals, and that allows coefficients to gradually
increase (decrease) until they reach a constant value (become zero). By
"stability", we mean that the number of misses and extras in the support
estimate remain bounded by time-invariant values (in turn implying a
time-invariant bound on LS-CS error). The concept is meaningful only if the
bounds are small compared to the support size. Numerical experiments backing
our claims are shown.Comment: Accepted (with mandatory minor revisions) to IEEE Trans. Signal
Processing. 12 pages, 5 figure
Norm minimized Scattering Data from Intensity Spectra
We apply the minimizing technique of compressive sensing (CS) to
non-linear quadratic observations. For the example of coherent X-ray scattering
we provide the formulae for a Kalman filter approach to quadratic CS and show
how to reconstruct the scattering data from their spatial intensity
distribution.Comment: 26 pages, 10 figures, reordered section
Active Classification for POMDPs: a Kalman-like State Estimator
The problem of state tracking with active observation control is considered
for a system modeled by a discrete-time, finite-state Markov chain observed
through conditionally Gaussian measurement vectors. The measurement model
statistics are shaped by the underlying state and an exogenous control input,
which influence the observations' quality. Exploiting an innovations approach,
an approximate minimum mean-squared error (MMSE) filter is derived to estimate
the Markov chain system state. To optimize the control strategy, the associated
mean-squared error is used as an optimization criterion in a partially
observable Markov decision process formulation. A stochastic dynamic
programming algorithm is proposed to solve for the optimal solution. To enhance
the quality of system state estimates, approximate MMSE smoothing estimators
are also derived. Finally, the performance of the proposed framework is
illustrated on the problem of physical activity detection in wireless body
sensing networks. The power of the proposed framework lies within its ability
to accommodate a broad spectrum of active classification applications including
sensor management for object classification and tracking, estimation of sparse
signals and radar scheduling.Comment: 38 pages, 6 figure
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