Thesis study-euclidean Distance matrices-theory,algorithms,applications (1)

Translated from Euclidean Distance matrices:essential theory, algorithms, and applications

The EDMS is the average distance matrix between points. The aim of this paper is to introduce the application of EMD in the field of signal processing and show how the EDM is used to design the algorithm-repairing and de-noising the distance data. At the same time, the position calibration of the microphone (microphone position calibration), ultrasonic tomography (Ultrosound tomography), the space reconstruction from Echoes and phase recovery applications.

Introduction

Suppose you have a Swiss train timetable but there is no map, but this is enough to rebuild a rough alpine map, even if the train time does not reflect the distance even part of the time is not known.

We often deal with distance data because they are easy to measure and evaluate. For example, in wireless sensor networks, sensor nodes can measure the intensity of signal packets transmitted by other nodes or the pulse arrival time (TOA) sent by their neighboring nodes, which is self-locating (self-localization).

Sometimes the data is not a matrix, but we can look for a matrix representation just like a psychometric test. In fact, psychological testing is the origin of many EDM-related methods, including multidimensional scale analysis (MDS), which uses points in multidimensional spaces to denote perceptual or psychological measurement relationships between different stimuli (Baidu).

The EDM is a good starting point for a useful description of point sets and algorithmic design. A classic task is to restore the original point settings: It only requires a symmetric matrix for eigenvalue decomposition (EVD). In fact, most of the Euclidean distance problem reconstruction point sets are accompanied by the following issues:

1) noise from the data

2) Some distance data is missing

3) Distance not recognized (unlabelled)

The distance geometry has two basic problems: one is to give a matrix, to determine whether it is an EDM, and the other is to give some incomplete distance data, the dimension of the smallest affine space under the known embedded dimension--to determine if there is a point configuration (configuration of points)

From PoIntS to EdMs and back

X is the d*n matrix,

The average distance between Xi and XJ is, and defines

||.|| Represents the Euclidean norm, i.e.

1 represents the column vector of the n*1, that is, the transpose of [1 1 1...1], and the 1 transpose is the line vector of the 1*n; the EDM (X) is actually

Similarly, to make, then

Formulas 3 and 4 reveal an important attribute: the rank of X is up to D, the rank of the most is D, and the other two addend rank 1 for Equation 3, thus getting: the rank of an EDM is up to d+2 (T1)

This theorem shows that the rank of the EDM is independent of the number of points, and in many applications, D is 3 or less whereas n can be thousands. According to theorem 1, the rank of such matrices is up to 5. The most important thing in Theorem 1 is the affine dimension of the point set. Any affine subspace is a transformation of a linear subspace, the important uniqueness (essential uniqueness). As shown, taking any point from all the points of an affine subspace can uniquely describe a parallel subspace with 0 vectors.

ESSENTIAL Uniqueness
When dealing with an inverse problem (inverse problem), we need to understand which ones can be re-obtained and which are lost. The rebuild point configuration usually increases its size (size), and the number of pairs of distances is much larger than the size of the coordinate description. It is obvious that the rigid transformation does not change the distance of the fixed point, as we learned from equations 3 and 4: EDM (X) =, so the algebraic method (flip, reflection) does not change the distance. So for the rotation point set (q is the orthogonal matrix of d*d, i.e.):

The translation matrix is obtained by d*1 's column vector b

So

This means that we cannot get the exact location of a point by distance data, and different reconstruction steps get different point configurations, as shown in:

Reconstruction the point SET from DISTANCE
Equation 3 gives us a way to compute the point configuration from the distance matrix.

Assuming that point X1 is the origin, the first column of D contains the square norm of the point vector, and D1 is the first column of D, which is

Because the diagonal element happens to be, the

Since G is a symmetric semi-positive definite matrix (PSD), where

So, for the rebuild point configuration,.

So, we have Theorem 2: When and only if the condition is satisfied with the S

is a PSD, D is an EDM.

It's easy to know, s=e1. The translation point set allows the X1 to be panned to the origin, through the right-multiplicative matrix of the EDM (x) left-multiplicative matrix, which we will eventually get, and the rebuild point configuration will have the first dot at the origin (with the original point at the origin).

On the other hand, s= (1/n) 1 o'clock, the origin of the coordinate system is the centroid of the point set, so the matrix is called the Geometric Center matrix

To illustrate this, we define the centroid of X as the average of all the point sets

Then subtract this amount from the point set, i.e.

Thesis study-euclidean Distance matrices-theory,algorithms,applications (1)