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.
%0 Generic
%1 dokmanic2015euclidean
%A Dokmanic, Ivan
%A Parhizkar, Reza
%A Ranieri, Juri
%A Vetterli, Martin
%D 2015
%K Euclidean-distance_matrices MDS dimensionality_reduction inverse_problems matrix_decomposition review visualization
%R 10.1109/MSP.2015.2398954
%T Euclidean Distance Matrices: Essential Theory, Algorithms and
Applications
%U http://arxiv.org/abs/1502.07541
%X 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.
@misc{dokmanic2015euclidean,
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.},
added-at = {2018-10-04T06:20:05.000+0200},
author = {Dokmanic, Ivan and Parhizkar, Reza and Ranieri, Juri and Vetterli, Martin},
biburl = {https://www.bibsonomy.org/bibtex/2098e4ad2e7f76d40fb797eedf5f80562/peter.ralph},
doi = {10.1109/MSP.2015.2398954},
interhash = {0c345b3f29bcc3449b788ff7efc945d2},
intrahash = {098e4ad2e7f76d40fb797eedf5f80562},
keywords = {Euclidean-distance_matrices MDS dimensionality_reduction inverse_problems matrix_decomposition review visualization},
note = {cite arxiv:1502.07541},
timestamp = {2018-10-04T06:20:05.000+0200},
title = {Euclidean Distance Matrices: Essential Theory, Algorithms and
Applications},
url = {http://arxiv.org/abs/1502.07541},
year = 2015
}