19 research outputs found
Bayesian Lower Bounds for Dense or Sparse (Outlier) Noise in the RMT Framework
Robust estimation is an important and timely research subject. In this paper,
we investigate performance lower bounds on the mean-square-error (MSE) of any
estimator for the Bayesian linear model, corrupted by a noise distributed
according to an i.i.d. Student's t-distribution. This class of prior
parametrized by its degree of freedom is relevant to modelize either dense or
sparse (accounting for outliers) noise. Using the hierarchical Normal-Gamma
representation of the Student's t-distribution, the Van Trees' Bayesian
Cram\'er-Rao bound (BCRB) on the amplitude parameters is derived. Furthermore,
the random matrix theory (RMT) framework is assumed, i.e., the number of
measurements and the number of unknown parameters grow jointly to infinity with
an asymptotic finite ratio. Using some powerful results from the RMT,
closed-form expressions of the BCRB are derived and studied. Finally, we
propose a framework to fairly compare two models corrupted by noises with
different degrees of freedom for a fixed common target signal-to-noise ratio
(SNR). In particular, we focus our effort on the comparison of the BCRBs
associated with two models corrupted by a sparse noise promoting outliers and a
dense (Gaussian) noise, respectively
Bayesian Estimation for Continuous-Time Sparse Stochastic Processes
We consider continuous-time sparse stochastic processes from which we have
only a finite number of noisy/noiseless samples. Our goal is to estimate the
noiseless samples (denoising) and the signal in-between (interpolation
problem).
By relying on tools from the theory of splines, we derive the joint a priori
distribution of the samples and show how this probability density function can
be factorized. The factorization enables us to tractably implement the maximum
a posteriori and minimum mean-square error (MMSE) criteria as two statistical
approaches for estimating the unknowns. We compare the derived statistical
methods with well-known techniques for the recovery of sparse signals, such as
the norm and Log (- relaxation) regularization
methods. The simulation results show that, under certain conditions, the
performance of the regularization techniques can be very close to that of the
MMSE estimator.Comment: To appear in IEEE TS
On the Continuity of Characteristic Functionals and Sparse Stochastic Modeling
The characteristic functional is the infinite-dimensional generalization of
the Fourier transform for measures on function spaces. It characterizes the
statistical law of the associated stochastic process in the same way as a
characteristic function specifies the probability distribution of its
corresponding random variable. Our goal in this work is to lay the foundations
of the innovation model, a (possibly) non-Gaussian probabilistic model for
sparse signals. This is achieved by using the characteristic functional to
specify sparse stochastic processes that are defined as linear transformations
of general continuous-domain white noises (also called innovation processes).
We prove the existence of a broad class of sparse processes by using the
Minlos-Bochner theorem. This requires a careful study of the regularity
properties, especially the boundedness in Lp-spaces, of the characteristic
functional of the innovations. We are especially interested in the functionals
that are only defined for p<1 since they appear to be associated with the
sparser kind of processes. Finally, we apply our main theorem of existence to
two specific subclasses of processes with specific invariance properties.Comment: 24 page
Generating Sparse Stochastic Processes Using Matched Splines
We provide an algorithm to generate trajectories of sparse stochastic
processes that are solutions of linear ordinary differential equations driven
by L\'evy white noises. A recent paper showed that these processes are limits
in law of generalized compound-Poisson processes. Based on this result, we
derive an off-the-grid algorithm that generates arbitrarily close
approximations of the target process. Our method relies on a B-spline
representation of generalized compound-Poisson processes. We illustrate
numerically the validity of our approach