3 research outputs found
Performance Bounds for Sparse Parametric Covariance Estimation in Gaussian Models
We consider estimation of a sparse parameter vector that determines the
covariance matrix of a Gaussian random vector via a sparse expansion into known
"basis matrices". Using the theory of reproducing kernel Hilbert spaces, we
derive lower bounds on the variance of estimators with a given mean function.
This includes unbiased estimation as a special case. We also present a
numerical comparison of our lower bounds with the variance of two standard
estimators (hard-thresholding estimator and maximum likelihood estimator)
The RKHS Approach to Minimum Variance Estimation Revisited: Variance Bounds, Sufficient Statistics, and Exponential Families
The mathematical theory of reproducing kernel Hilbert spaces (RKHS) provides
powerful tools for minimum variance estimation (MVE) problems. Here, we extend
the classical RKHS based analysis of MVE in several directions. We develop a
geometric formulation of five known lower bounds on the estimator variance
(Barankin bound, Cramer-Rao bound, constrained Cramer-Rao bound, Bhattacharyya
bound, and Hammersley-Chapman-Robbins bound) in terms of orthogonal projections
onto a subspace of the RKHS associated with a given MVE problem. We show that,
under mild conditions, the Barankin bound (the tightest possible lower bound on
the estimator variance) is a lower semicontinuous function of the parameter
vector. We also show that the RKHS associated with an MVE problem remains
unchanged if the observation is replaced by a sufficient statistic. Finally,
for MVE problems conforming to an exponential family of distributions, we
derive novel closed-form lower bound on the estimator variance and show that a
reduction of the parameter set leaves the minimum achievable variance
unchanged
An RKHS Approach to Estimation with Sparsity Constraints
The investigation of the effects of sparsity or sparsity constraints in
signal processing problems has received considerable attention recently.
Sparsity constraints refer to the a priori information that the object or
signal of interest can be represented by using only few elements of a
predefined dictionary. Within this thesis, sparsity refers to the fact that a
vector to be estimated has only few nonzero entries. One specific field
concerned with sparsity constraints has become popular under the name
Compressed Sensing (CS). Within CS, the sparsity is exploited in order to
perform (nearly) lossless compression. Moreover, this compression is carried
out jointly or simultaneously with the process of sensing a physical quantity.
In contrast to CS, one can alternatively use sparsity to enhance signal
processing methods. Obviously, sparsity constraints can only improve the
obtainable estimation performance since the constraints can be interpreted as
an additional prior information about the unknown parameter vector which is to
be estimated. Our main focus will be on this aspect of sparsity, i.e., we
analyze how much we can gain in estimation performance due to the sparsity
constraints.Comment: PHD Thesi