10 research outputs found
Information Geometric Approach to Bayesian Lower Error Bounds
Information geometry describes a framework where probability densities can be
viewed as differential geometry structures. This approach has shown that the
geometry in the space of probability distributions that are parameterized by
their covariance matrix is linked to the fundamentals concepts of estimation
theory. In particular, prior work proposes a Riemannian metric - the distance
between the parameterized probability distributions - that is equivalent to the
Fisher Information Matrix, and helpful in obtaining the deterministic
Cram\'{e}r-Rao lower bound (CRLB). Recent work in this framework has led to
establishing links with several practical applications. However, classical CRLB
is useful only for unbiased estimators and inaccurately predicts the mean
square error in low signal-to-noise (SNR) scenarios. In this paper, we propose
a general Riemannian metric that, at once, is used to obtain both Bayesian CRLB
and deterministic CRLB along with their vector parameter extensions. We also
extend our results to the Barankin bound, thereby enhancing their applicability
to low SNR situations.Comment: 5 page
Direction of arrival estimation based on information geometry
In this paper, a new direction of arrival (DOA) estimation approach is devised using concepts from information geometry (IG). The proposed method uses geodesic distances in the statistical manifold of probability distributions parametrized by their covariance matrix to estimate the direction of arrival of several sources. In order to obtain a practical method, the DOA estimation is treated as a single-variable optimization problem, for which the DOA solutions are found by means of a line search. The relation between the proposed method and MVDR beamformer is elucidated. An evaluation of its performance is carried out by means of Monte Carlo simulations and it is shown that the proposed method provides improved resolution capabilities at low SNR with respect to MUSIC and MVDR.Accepted Author ManuscriptCircuits and System
A Framework for Population-Based Stochastic Optimization on Abstract Riemannian Manifolds
We present Extended Riemannian Stochastic Derivative-Free Optimization
(Extended RSDFO), a novel population-based stochastic optimization algorithm on
Riemannian manifolds that addresses the locality and implicit assumptions of
manifold optimization in the literature.
We begin by investigating the Information Geometrical structure of
statistical model over Riemannian manifolds. This establishes a geometrical
framework of Extended RSDFO using both the statistical geometry of the decision
space and the Riemannian geometry of the search space. We construct locally
inherited probability distribution via an orientation-preserving diffeomorphic
bundle morphism, and then extend the information geometrical structure to
mixture densities over totally bounded subsets of manifolds. The former relates
the information geometry of the decision space and the local point estimations
on the search space manifold. The latter overcomes the locality of parametric
probability distributions on Riemannian manifolds.
We then construct Extended RSDFO and study its structure and properties from
a geometrical perspective. We show that Extended RSDFO's expected fitness
improves monotonically and it's global eventual convergence in finitely many
steps on connected compact Riemannian manifolds.
Extended RSDFO is compared to state-of-the-art manifold optimization
algorithms on multi-modal optimization problems over a variety of manifolds.
In particular, we perform a novel synthetic experiment on Jacob's ladder to
motivate and necessitate manifold optimization. Jacob's ladder is a non-compact
manifold of countably infinite genus, which cannot be expressed as polynomial
constraints and does not have a global representation in an ambient Euclidean
space. Optimization problems on Jacob's ladder thus cannot be addressed by
traditional (constraint) optimization methods on Euclidean spaces.Comment: The present abstract is slightly altered from the PDF version due to
the limitation "The abstract field cannot be longer than 1,920 characters