Jose A. Costa

Distributed weighted-multidimensional scaling for node localization in sensor networks

Accurate, distributed localization algorithms are needed for a wide variety of wireless sensor network applications. This article introduces a scalable, distributed weighted-multidimensional scaling (dwMDS) algorithm that adaptively emphasizes the most accurate range measurements and naturally accounts for communication constraints within the sensor network. Each node adaptively chooses a neighborhood of sensors, updates its position estimate by minimizing a local cost function and then passes this update to neighboring sensors. Derived bounds on communication requirements provide insight on the energy efficiency of the proposed distributed method versus a centralized approach. For received signal-strength (RSS) based range measurements, we demonstrate via simulation that location estimates are nearly unbiased with variance close to the Cramér-Rao lower bound. Further, RSS and time-of-arrival (TOA) channel measurements are used to demonstrate performance as good as the centralized maximum-likelihood estimator (MLE) in a real-world sensor network.

Geodesic entropic graphs for dimension and entropy estimation in manifold learning

In the manifold learning problem, one seeks to discover a smooth low dimensional surface, i.e., a manifold embedded in a higher dimensional linear vector space, based on a set of measured sample points on the surface. In this paper, we consider the closely related problem of estimating the manifolds intrinsic dimension and the intrinsic entropy of the sample points. Specifically, we view the sample points as realizations of an unknown multivariate density supported on an unknown smooth manifold. We introduce a novel geometric approach based on entropic graph methods. Although the theory presented applies to this general class of graphs, we focus on the geodesic-minimal-spanning-tree (GMST) to obtaining asymptotically consistent estimates of the manifold dimension and the Re/spl acute/nyi /spl alpha/-entropy of the sample density on the manifold. The GMST approach is striking in its simplicity and does not require reconstruction of the manifold or estimation of the multivariate density of the samples. The GMST method simply constructs a minimal spanning tree (MST) sequence using a geodesic edge matrix and uses the overall lengths of the MSTs to simultaneously estimate manifold dimension and entropy. We illustrate the GMST approach on standard synthetic manifolds as well as on real data sets consisting of images of faces.

Achieving high-accuracy distributed localization in sensor networks

Accurate, distributed localization algorithms are needed for a wide variety of wireless sensor network applications. This paper introduces a scalable, distributed weighted-multidimensional scaling (dwMDS) algorithm that adaptively emphasizes the most accurate range measurements available and naturally accounts for communication constraints within the sensor network. For received signal-strength (RSS) based range measurements, we demonstrate via simulation that location estimates are nearly unbiased with variance close to the Cramer-Rao lower bound (CRB). Further, RSS and time-of-arrival (TOA) channel measurements are used to demonstrate performance as good as the centralized maximum-likelihood estimator (MLE) in a real-world sensor network.

Estimating Local Intrinsic Dimension with k-Nearest Neighbor Graphs

Many high-dimensional data sets of practical interest exhibit a varying complexity in different parts of the data space. This is the case, for example, of databases of images containing many samples of a few textures of different complexity. Such phenomena can he modeled by assuming that the data lies on a collection of manifolds with different intrinsic dimensionalities. In this extended abstract, we introduce a method to estimate the local dimensionality associated with each point in a data set, without any prior information about the manifolds, their quantity and their sampling distributions. The proposed method uses a global dimensionality estimator based on k-nearest neighbor (k-NN) graphs, together with an algorithm for computing neighborhoods in the data with similar topological properties

Classification constrained dimensionality reduction

In this paper, we propose a nonlinear dimensionality reduction method aimed at extracting lower-dimensional features relevant for classification tasks. This is obtained by modifying the Laplacian approach to manifold learning through the introduction of class dependent constraints. Using synthetic data sets, we show that the proposed algorithm can greatly improve both supervised and semi-supervised learning problems.

Determining Intrinsic Dimension and Entropy of High-Dimensional Shape Spaces

Given a finite set of random samples from a smooth Riemannian manifold embedded in ℝd, two important questions are: what is the intrinsic dimension of the manifold and what is the entropy of the underlying sampling distribution on the manifold? These questions naturally arise in the study of shape spaces generated by images or signals for the purposes of shape classification, shape compression, and shape reconstruction. This chapter is concerned with two simple estimators of dimension and entropy based on the lengths of the geodesic minimal spanning tree (GMST) and the k-nearest neighbor (k-NN) graph. We provide proofs of strong consistency of these estimators under weak assumptions of compactness of the manifold and boundedness of the Lebesgue sampling density supported on the manifold. We illustrate these estimators on the MNIST database of handwritten digits.

Learning Sensor Location from Signal Strength and Connectivity

Received signal strength (RSS) or connectivity, i.e., whether or not two devices can communicate, are two relatively inexpensive (in terms of device and energy costs) measurements at the receiver that indicate the distance from the transmitter. Such measurements can either be quickly dismissed as too unreliable for localization, or idealized by ignoring the non-circular nature of a transmitter’s coverage area. This chapter finds a middle ground between these two extremes by using measurement-based statistical models to represent the inaccuracies of RSS and connectivity.

On Solutions to Multivariate Maximum α-Entropy Problems

Entropy has been widely employed as an optimization function for problems in computer vision and pattern recognition. To gain insight into such methods it is important to characterize the behavior of the maximum-entropy probability distributions that result from the entropy optimization. The aim of this paper is to establish properties of multivariate distributions maximizing entropy for a general class of entropy functions, called Renyi’s α-entropy, under a covariance constraint. First we show that these entropy-maximizing distributions exhibit interesting properties, such as spherical invariance, and have a stochastic Gaussian-Gamma mixture representation. We then turn to the question of stability of the class of entropy-maximizing distributions under addition.

Entropic graphs for manifold learning

We propose a new algorithm that simultaneously estimates the intrinsic dimension and intrinsic entropy of random data sets lying on smooth manifolds. The method is based on asymptotic properties of entropic graph constructions. In particular, we compute the Euclidean k-nearest neighbors (k-NN) graph over the sample points and use its overall total edge length to estimate intrinsic dimension and entropy. The algorithm is validated on standard synthetic manifolds.

A characterization of the multivariate distributions maximizing Renyi entropy

We characterize the multivariate probability distributions that maximize the Renyi entropy under covariance constraint. Then, we show that these distributions are stable under a particular type of convolution.