Single Element Error Correction/ in a Euclidean Distance Matrix

TL;DR

This paper develops three divide-and-conquer algorithms for exact error correction in noisy Euclidean distance matrices, capable of handling large datasets efficiently and providing theoretical guarantees for identifying corrupted elements.

Contribution

It introduces novel algorithms combining facial reduction and Gale transforms for pinpointing single corrupted distances in EDMs, with proven correctness and scalability.

Findings

Successfully corrected large EDMs with 100,000 nodes in about one minute.

Provided a simple characterization for identifying corrupted elements in 2D EDMs.

Showed when the nearest EDM approach correctly identifies the corrupted element.

Abstract

We consider the \emph{exact} error correction of a noisy Euclidean distance matrix, EDM, where the elements are the squared distances between $n$ points in $R^d$. For our problem we are given two facts: (i) the embedding dimension, $d$, (ii) \emph{exactly one} distance in the data is corrupted by \emph{nonzero noise}. But we do \underline{not} know the magnitude nor position of the noise. Thus there is a combinatorial element to the problem. We present three solution techniques. These use three divide and conquer strategies in combination with three versions of facial reduction that use: exposing vectors, facial vectors, and Gale transforms. This sheds light on the connections between the various forms of facial reduction related to Gale transforms. Our highly successful empirics confirm the success of these approaches as we can solve huge problems of the order of $100,000$ nodes in approximately one minute to machine precision. \\Our algorithm depends on identifying whether a principal submatrix of the \EDM contains the corrupted element. We provide a theorem for doing this that is related to the existing results for identifying \emph{yielding} elements, i.e.,~we provide a characterization for guaranteeing the perturbed EDM remains an EDM with embedding dimension $d$. The characterization is particularly simple in the $d=2$ case. \\In addition, we characterize when the intuitive approach of the nearest EDM problem, solves our problem. In fact, we show that this happens if, and only if, the original distance element is $0$, degenerate, and the perturbation is negative.