2 research outputs found
Entropic and displacement interpolation: a computational approach using the Hilbert metric
Monge-Kantorovich optimal mass transport (OMT) provides a blueprint for
geometries in the space of positive densities -- it quantifies the cost of
transporting a mass distribution into another. In particular, it provides
natural options for interpolation of distributions (displacement interpolation)
and for modeling flows. As such it has been the cornerstone of recent
developments in physics, probability theory, image processing, time-series
analysis, and several other fields. In spite of extensive work and theoretical
developments, the computation of OMT for large scale problems has remained a
challenging task. An alternative framework for interpolating distributions,
rooted in statistical mechanics and large deviations, is that of Schroedinger
bridges (entropic interpolation). This may be seen as a stochastic
regularization of OMT and can be cast as the stochastic control problem of
steering the probability density of the state-vector of a dynamical system
between two marginals. In this approach, however, the actual computation of
flows had hardly received any attention. In recent work on Schroedinger bridges
for Markov chains and quantum evolutions, we noted that the solution can be
efficiently obtained from the fixed-point of a map which is contractive in the
Hilbert metric. Thus, the purpose of this paper is to show that a similar
approach can be taken in the context of diffusion processes which i) leads to a
new proof of a classical result on Schroedinger bridges and ii) provides an
efficient computational scheme for both, Schroedinger bridges and OMT. We
illustrate this new computational approach by obtaining interpolation of
densities in representative examples such as interpolation of images.Comment: 20 pages, 7 figure
Novel Results on the Factorization and Estimation of Spectral Densities
This dissertation is divided into two main parts. The first part is concerned with one of the most classical and central problems in Systems and Control Theory, namely the factorization of rational matrix-valued spectral densities, commonly known as the spectral factorization problem. Spectral factorization is a fundamental tool for the solution of a variety of problems involving second-order statistics and quadratic cost functions in control, estimation, signal processing and communications. It can be thought of as the frequency-domain counterpart of the ubiquitous Algebraic Riccati Equation and it is intimately connected with the celebrated Kálmán-Yakubovich-Popov Lemma and, therefore, to passivity theory. Here, we provide a rather in-depth and comprehensive analysis of this problem in the discrete-time setting, a scenario which is becoming increasingly pervasive in control applications. The starting point in our analysis is a general spectral factorization result in the same vein of Dante C. Youla. Building on this fundamental result, we then investigate some key issues related to minimality and parametrization of minimal spectral factors of a given spectral density. To conclude, we show how to extend some of the ideas and results to the more general indefinite or J-spectral factorization problem, a technique of paramount importance in robust control and estimation theory.
In the second part of the dissertation, we consider the problem of estimating a spectral density from a finite set of measurements. Following the Byrnes-Georgiou-Lindquist THREE (Tunable High REsolution Estimation) paradigm, we look at spectral estimation as an optimization problem subjected to a generalized moment constraint. In this framework, we examine the global convergence of an efficient algorithm for the estimation of scalar spectral densities that hinges on the Kullback-Leibler criterion. We then move to the multivariate setting by addressing the delicate issue of existence of solutions to a parametric spectral estimation problem. Eventually, we study the geometry of the space of spectral densities by revisiting two natural distances defined in cones for the case of rational spectra. These new distances are used to formulate a "robust" version of THREE-like spectral estimation