Euclidean distance matrices (EDMs) are matrices of the squared distances between points. The definition is deceivingly simple; thanks to their many useful properties, they have found applications in psychometrics, crystallography, machine learning, wireless sensor networks, acoustics, and more. Despite the usefulness of EDMs, they seem to be insufficiently known in the signal processing community. Our goal is to rectify this mishap in a concise tutorial. We review the fundamental properties of EDMs, such as rank or (non)definiteness, and show how the various EDM properties can be used to design algorithms for completing and denoising distance data. Along the way, we demonstrate applications to microphone position calibration, ultrasound tomography, room reconstruction from echoes, and phase retrieval. By spelling out the essential algorithms, we hope to fast-track the readers in applying EDMs to their own problems. The code for all of the described algorithms and to generate the figures in the article is available online at http://lcav.epfl.ch/ivan.dokmanic. Finally, we suggest directions for further research. IntroductIonImagine that you land at Geneva International Airport with the Swiss train schedule but no map. Perhaps surprisingly, this may be sufficient to reconstruct a rough (or not so rough) map of the Alpine country, even if the train times poorly translate to distances or if some of the times are unknown. The way to do it is by using EDMs; for an example, see "Swiss Trains (Swiss Map Reconstruction)."We often work with distances because they are convenient to measure or estimate. In wireless sensor networks, for example, the sensor nodes measure the received signal strengths of the packets sent by other nodes or the time of arrival (TOA) of pulses emitted by their neighbors [1]. Both of these proxies allow for distance estimation between pairs of nodes; thus, we can attempt to reconstruct the network topology. This is often termed self-localization [2]- [4]. The molecular conformation problem is another instance of a distance problem [5], and so is reconstructing a room's geometry from echoes [6]. Less obviously, sparse phase retrieval [7] can be converted to a distance problem and addressed using EDMs.Sometimes the data are not metric, but we seek a metric representation, as it happens commonly in psychometrics [8]. As a matter of fact, the psychometrics community is at the root of the development of a number of tools related to EDMs, including multidimensional scaling (MDS)-the problem of finding the best point set representation of a given set of distances. More abstractly, we can study EDMs for objects such as images, which live in highdimensional vector spaces [9].EDMs are a useful description of the point sets and a starting point for algorithm design. A typical task is to retrieve the original point configuration: it may initially come as a surprise that this requires no more than an eigenvalue decomposition (EVD) of a symmetric matrix. In fact, the majority of Euclidean distance problems requi...

A classic problem is the estimation of a set of parameters from measurements collected by only a few sensors. The number of sensors is often limited by physical or economical constraints and their placement is of fundamental importance to obtain accurate estimates. Unfortunately, the selection of the optimal sensor locations is intrinsically combinatorial and the available approximation algorithms are not guaranteed to generate good solutions in all cases of interest. We propose FrameSense, a greedy algorithm for the selection of optimal sensor locations. The core cost function of the algorithm is the frame potential, a scalar property of matrices that measures the orthogonality of its rows. Notably, FrameSense is the first algorithm that is near-optimal in terms of mean square error, meaning that its solution is always guaranteed to be close to the optimal one. Moreover, we show with an extensive set of numerical experiments that FrameSense achieves state-of-the-art performance while having the lowest computational cost, when compared to other greedy methods.Comment: 13 pages, accepted for publication on IEEE TS

We study the spatiotemporal sampling of a diffusion field generated by K point sources, aiming to fully reconstruct the unknown initial field distribution from the sample measurements. The sampling operator in our problem can be described by a matrix derived from the diffusion model. We analyze the important properties of the sampling matrices, leading to precise bounds on the spatial and temporal sampling densities under which perfect field reconstruction is feasible. Moreover, our analysis indicates that it is possible to compensate linearly for insufficient spatial sampling densities by oversampling in time. Numerical simulations on initial field reconstruction under different spatiotemporal sampling densities confirm our theoretical results.

Abstract-We consider the problem of reconstructing a diffusion field, such as temperature, from samples collected by a sensor network. Motivated by the fast decay of the eigenvalues of the diffusion equation, we approximate the field by a truncated series. We show that the approximation error decays rapidly with time. On the other hand, the information content in the field also decays with time, suggesting the need for a proper choice of the sampling strategy. We propose two algorithms for sampling and reconstruction of the field. The first one reconstructs the distribution of point sources appearing at known times using the finite rate of innovation (FRI) framework. The second algorithm addresses a more difficult problem of estimating the unknown times at which the point sources appear, in addition to their locations and magnitudes. It relies on the assumption that the sources appear at distinct times. We verify that the algorithms are capable of reconstructing the field accurately through a set of numerical experiments. Specifically, we show that the second algorithm successfully recovers an arbitrary number of sources with unknown release times, satisfying the assumption. For simplicity, we develop the 1-D theory, noting the possibility of extending the framework to more general domains.

Abstract-Wireless sensor networks are often designed to perform two tasks: sensing a physical field and transmitting the data to end-users. A crucial design aspect of a WSN is the minimization of the overall energy consumption. Previous researchers aim at optimizing the energy spent for the communication, while mostly ignoring the energy cost of sensing. Recently, it has been shown that considering the sensing energy cost can be beneficial for further improving the overall energy efficiency. More precisely, sparse sensing techniques were proposed to reduce the amount of collected samples and recover the missing data using data statistics. While the majority of these techniques use fixed or random sampling patterns, we propose adaptively learning the signal model from the measurements and using the model to schedule when and where to sample the physical field. The proposed method requires minimal on-board computation, no inter-node communications, and achieves appealing reconstruction performance. With experiments on real-world datasets, we demonstrate significant improvements over both traditional sensing schemes and the state-of-the-art sparse sensing schemes, particularly when the measured data is characterized by a strong intra-sensor (temporal) or inter-sensors (spatial) correlation.Index Terms-Wireless sensor networks, sparse sensing, adaptive sampling scheduling, compressive sensing, energy efficiency.