Doron Blatt

Energy-based sensor network source localization via projection onto convex sets

This correspondence addresses the problem of locating an acoustic source using a sensor network in a distributed manner, i.e., without transmitting the full data set to a central point for processing. This problem has been traditionally addressed through the maximum-likelihood framework or nonlinear least squares. These methods, even though asymptotically optimal under certain conditions, pose a difficult global optimization problem. It is shown that the associated objective function may have multiple local optima and saddle points, and hence any local search method might stagnate at a suboptimal solution. In this correspondence, we formulate the problem as a convex feasibility problem and apply a distributed version of the projection-onto-convex-sets (POCS) method. We give a closed-form expression for the projection phase, which usually constitutes the heaviest computational aspect of POCS. Conditions are given under which, when the number of samples increases to infinity or in the absence of measurement noise, the convex feasibility problem has a unique solution at the true source location. In general, the method converges to a limit point or a limit cycle in the neighborhood of the true location. Simulation results show convergence to the global optimum with extremely fast convergence rates compared to the previous methods

A Convergent Incremental Gradient Method with a Constant Step Size

An incremental aggregated gradient method for minimizing a sum of continuously differentiable functions is presented. The method requires a single gradient evaluation per iteration and uses a constant step size. For the case that the gradient is bounded and Lipschitz continuous, we show that the method visits infinitely often regions in which the gradient is small. Under certain unimodality assumptions, global convergence is established. In the quadratic case, a global linear rate of convergence is shown. The method is applied to distributed optimization problems arising in wireless sensor networks, and numerical experiments compare the new method with other incremental gradient methods.

Sensor network source localization via projection onto convex sets (POCS)

This correspondence addresses the problem of locating an acoustic source using a sensor network in a distributed manner, i.e., without transmitting the full data set to a central point for processing. This problem has been traditionally addressed through the maximum-likelihood framework or nonlinear least squares. These methods, even though asymptotically optimal under certain conditions, pose a difficult global optimization problem. It is shown that the associated objective function may have multiple local optima and saddle points, and hence any local search method might stagnate at a suboptimal solution. In this correspondence, we formulate the problem as a convex feasibility problem and apply a distributed version of the projection-onto-convex-sets (POCS) method. We give a closed-form expression for the projection phase, which usually constitutes the heaviest computational aspect of POCS. Conditions are given under which, when the number of samples increases to infinity or in the absence of measurement noise, the convex feasibility problem has a unique solution at the true source location. In general, the method converges to a limit point or a limit cycle in the neighborhood of the true location. Simulation results show convergence to the global optimum with extremely fast convergence rates compared to the previous methods

Distributed maximum likelihood estimation for sensor networks

The problem of finding the maximum likelihood estimator of a commonly observed model, based on data collected by a sensor network under power and bandwidth constraints, is considered. In particular, a case where the sensors cannot fully share their data is treated. An iterative algorithm that relaxes the requirement of sharing all the data is given. The algorithm is based on a local Fisher scoring method and an iterative information sharing procedure. The case where the sensors share sub-optimal estimates is also analyzed. The asymptotic distribution of the estimates is derived and used to provide a means of discrimination between estimates that are associated with different local maxima of the log-likelihood function. The results are validated by a simulation.

Adaptive multi-modality sensor scheduling for detection and tracking of smart targets

This paper considers the problem of sensor scheduling for the purposes of detection and tracking of “smart” targets. Smart targets are targets that can detect when they are under surveillance and react in a manner that makes future surveillance more difficult. We take a reinforcement learning approach to adaptively schedule a multi-modality sensor so as to most quickly and effectively detect the presence of smart targets and track them as they travel through a surveillance region. An optimal scheduling strategy, which would simultaneously address the issue of target detection and tracking, is very challenging computationally. To avoid this difficulty, we use a two stage approach where targets are first detected and then handed off to a tracking algorithm. We investigate algorithms capable of choosing whether to use the active or passive mode of an agile sensor. The active mode is easily detected by the target, which makes the target prefer to move into hide mode. The passive mode is nearly undetectable to the target. However, the active mode has substantially better detection and tracking capabilities then the passive mode. Using this setup, we characterize the advantage of a non-myopic policy with respect to myopic and random polices for multitarget detection and tracking.

APOCS: a rapidly convergent source localization algorithm for sensor networks

This paper addresses the problem of locating an acoustic source using a sensor network in a distributed manner, i.e., without transmitting the full data set to a central point for processing. This problem has been traditionally addressed through the maximum likelihood framework or nonlinear least squares. These methods, even though asymptotically optimal under certain conditions, pose a difficult global optimization problem. It is shown that the associated objective function may have multiple local optima and hence local search methods might stagnate at a sub-optimal solution. In this paper, we treat the problem in its convex feasibility formulation. We propose the aggregated projection onto convex sets (APOCS) method, which, in contrast to the original POCS method, converges to a meaningful limit even when the problem is infeasible without requiring a diminishing step size. Simulation results show convergence to the global optimum with significantly faster convergence rates compared to the previous methods

On Tests for Global Maximum of the Log-Likelihood Function

Given the location of a relative maximum of the log-likelihood function, how to assess whether it is the global maximum? This paper investigates an existing statistical tool, which, based on asymptotic analysis, answers this question by posing it as a hypothesis testing problem. A general framework for constructing tests for global maximum is given. The characteristics of the tests are investigated for two cases: correctly specified model and model mismatch. A finite sample approximation to the power is given, which gives a tool for performance prediction and a measure for comparison between tests. The sensitivity of the tests to model mismatch is analyzed in terms of the Renyi divergence and the Kullback-Leibler divergence between the true underlying distribution and the assumed parametric class and tests that are insensitive to small deviations from the model are derived thereby overcoming a fundamental weakness of existing tests. The tests are illustrated for three applications: passive localization or direction finding using an array of sensors, estimating the parameters of a Gaussian mixture model, and estimation of superimposed exponentials in noise-problems that are known to suffer from local maxima.

Incremental optimization transfer algorithms: application to transmission tomography

No convergent ordered subsets (OS) type image reconstruction algorithms for transmission tomography have been proposed to date. In contrast, in emission tomography, there are two known families of convergent OS algorithms: methods that use relaxation parameters (Ahn and Fessler, 2003), and methods based on the incremental expectation maximization (EM) approach (Hsiao et al., 2002). This paper generalizes the incremental EM approach by introducing a general framework that we call ¿incremental optimization transfer.¿ Like incremental EM methods, the proposed algorithms accelerate convergence speeds and ensure global convergence (to a stationary point) under mild regularity conditions without requiring inconvenient relaxation parameters. The general optimization transfer framework enables the use of a very broad family of non-EM surrogate functions. In particular, this paper provides the first convergent OS-type algorithm for transmission tomography. The general approach is applicable to both monoenergetic and polyenergetic transmission scans as well as to other image reconstruction problems. We propose a particular incremental optimization transfer method for (nonconcave) penalized-likelihood (PL) transmission image reconstruction by using separable paraboloidal surrogates (SPS). Results show that the new ¿transmission incremental optimization transfer (TRIOT)¿ algorithm is faster than nonincremental ordinary SPS and even OS-SPS yet is convergent.

Asymptotic Characterization of Log-Likelihood Maximization Based Algorithms and Applications

The asymptotic distribution of estimates that are based on a sub-optimal search for the maximum of the log-likelihood function is considered. In particular, estimation schemes that are based on a two-stage approach, in which an initial estimate is used as the starting point of a subsequent local maximization, are analyzed. We show that asymptotically the local estimates follow a Gaussian mixture distribution, where the mixture components correspond to the modes of the likelihood function. The analysis is relevant for cases where the log-likelihood function is known to have local maxima in addition to the global maximum, and there is no available method that is guaranteed to provide an estimate within the attraction region of the global maximum. Two applications of the analytic results are offered. The first application is an algorithm for finding the maximum likelihood estimator. The algorithm is best suited for scenarios in which the likelihood equations do not have a closed form solution, the iterative search is computationally cumbersome and highly dependent on the data length, and there is a risk of convergence to a local maximum. The second application is a scheme for aggregation of local estimates, e.g. generated by a network of sensors, at a fusion center. This scheme provides the means to intelligently combine estimates from remote sensors, where bandwidth constraints do not allow access to the complete set of data. The result on the asymptotic distribution is validated and the performance of the proposed algorithms is evaluated by computer simulations.

Tests for global maximum of the likelihood function

Given a relative maximum of the log-likelihood function, how to assess whether it is the global maximum? This paper investigates a statistical tool, which answers this question by posing it as a hypothesis testing problem. A general framework for constructing tests for the global maximum is given. The characteristics of the tests are investigated for two cases: correctly specified model and model mismatch. A finite sample approximation to the power is given, which gives a tool for performance prediction and a measure for comparison between tests. The tests are illustrated for two applications: estimating the parameters of a Gaussian mixture model and direction finding using an array of sensors - practical problems that are known to suffer from local maxima.