Euclidean Distance Matrices: Essential theory, algorithms, and applications


Euclidean distance matrices (EDMs) are matrices of the squared distances between points. The definition is deceivingly simple; because of their several useful properties, they need found applications in psychometrics, crystallography, machine learning, wireless sensor networks, acoustics, and more. Despite the usefulness of EDMs, they appear to be insufficiently known within the signal processing community. Our goal is to rectify this mishap in an exceedingly concise tutorial. We review the basic properties of EDMs, like rank or (non)definiteness, and show how the varied EDM properties can be used to design algorithms for completing and denoising distance knowledge. Along the means, we demonstrate applications to microphone position calibration, ultrasound tomography, area reconstruction from echoes, and part retrieval. By spelling out the essential algorithms, we have a tendency to hope to quick-track the readers in applying EDMs to their own issues. The code for all of the described algorithms and to generate the figures in the article is offered online at Finally, we suggest directions for any analysis.

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