Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userEuclidean Distance Matrices: Essential Theory, Algorithms and Applications
389
0
0.0
(
0
)
Added by
Ivan Dokmanic
Publication date
2015
fields
Informatics Engineering
and research's language is
English
Ask ChatGPT about the research
No Arabic abstract
Euclidean distance matrices (EDM) are matrices of 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. We show how 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. Matlab code for all the described algorithms, and to generate the figures in the paper, is available online. Finally, we suggest directions for further research.
rate research
Read More
Euclidean distance matrices (EDMs) are a major tool for localization from distances, with applications ranging from protein structure determination to global positioning and manifold learning. They are, however, static objects which serve to localize points from a snapshot of distances. If the objects move, one expects to do better by modeling the motion. In this paper, we introduce Kinetic Euclidean Distance Matrices (KEDMs)---a new kind of time-dependent distance matrices that incorporate motion. The entries of KEDMs become functions of time, the squared time-varying distances. We study two smooth trajectory models---polynomial and bandlimited trajectories---and show that these trajectories can be reconstructed from incomplete, noisy distance observations, scattered over multiple time instants. Our main contribution is a semidefinite relaxation (SDR), inspired by SDRs for static EDMs. Similarly to the static case, the SDR is followed by a spectral factorization step; however, because spectral factorization of polynomial matrices is more challenging than for constant matrices, we propose a new factorization method that uses anchor measurements. Extensive numerical experiments show that KEDMs and the new semidefinite relaxation accurately reconstruct trajectories from noisy, incomplete distance data and that, in fact, motion improves rather than degrades localization if properly modeled. This makes KEDMs a promising tool for problems in geometry of dynamic points sets.
In this paper, we present an acoustic localization system for multiple devices. In contrast to systems which localise a device relative to one or several anchor points, we focus on the joint localisation of several devices relative to each other. We present a prototype of our system on off-the-shelf smartphones. No user interaction is required, the phones emit acoustic pulses according to a precomputed schedule. Using the elapsed time between two times of arrivals (ETOA) method with sample counting, distances between the devices are estimated. These, possibly incomplete, distances are the input to an efficient and robust multi-dimensional scaling algorithm returning a position for each phone. We evaluated our system in real-world scenarios, achieving error margins of 15 cm in an office environment.
We show that Boolean matrix multiplication, computed as a sum of products of column vectors with row vectors, is essentially the same as Warshalls algorithm for computing the transitive closure matrix of a graph from its adjacency matrix. Warshalls algorithm can be generalized to Floyds algorithm for computing the distance matrix of a graph with weighted edges. We will generalize Boolean matrices in the same way, keeping matrix multiplication essentially equivalent to the Floyd-Warshall algorithm. This way, we get matrices over a semiring, which are similar to the so-called funny matrices. We discuss our implementation of operations on Boolean matrices and on their generalization, which make use of vector instructions.
Capturing high-dimensional (HD) data is a long-term challenge in signal processing and related fields. Snapshot compressive imaging (SCI) uses a two-dimensional (2D) detector to capture HD ($ge3$D) data in a {em snapshot} measurement. Via novel optical designs, the 2D detector samples the HD data in a {em compressive} manner; following this, algorithms are employed to reconstruct the desired HD data-cube. SCI has been used in hyperspectral imaging, video, holography, tomography, focal depth imaging, polarization imaging, microscopy, etc.~Though the hardware has been investigated for more than a decade, the theoretical guarantees have only recently been derived. Inspired by deep learning, various deep neural networks have also been developed to reconstruct the HD data-cube in spectral SCI and video SCI. This article reviews recent advances in SCI hardware, theory and algorithms, including both optimization-based and deep-learning-based algorithms. Diverse applications and the outlook of SCI are also discussed.
This tutorial aims to introduce the fundamentals of adversarial robustness of deep learning, presenting a well-structured review of up-to-date techniques to assess the vulnerability of various types of deep learning models to adversarial examples. This tutorial will particularly highlight state-of-the-art techniques in adversarial attacks and robustness verification of deep neural networks (DNNs). We will also introduce some effective countermeasures to improve the robustness of deep learning models, with a particular focus on adversarial training. We aim to provide a comprehensive overall picture about this emerging direction and enable the community to be aware of the urgency and importance of designing robust deep learning models in safety-critical data analytical applications, ultimately enabling the end-users to trust deep learning classifiers. We will also summarize potential research directions concerning the adversarial robustness of deep learning, and its potential benefits to enable accountable and trustworthy deep learning-based data analytical systems and applications.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
