Abstract
In this chapter , we consider the DGP on a very specific class of graphs: the (K + 1)-cliques, i.e., complete graphs on K + 1 vertices, where K is the dimension of the embedding space \(\mathbb {R}^{K}\).
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Notes
- 1.
Here is a setup where this might happen in practice. If the linear system \(Ay=b\) arises from taking equation differences \(\forall 1<h\le K\;({[h]}-{[1]})\), which is different but equivalent to Eq. (3.6), and \(x_1\) is arbitrarily set to 0, then an application of RealizeClique (Algorithm 2) would yield \(x_h\) with the last \(K-h+1\) components set to zero, for each \(h\le K\), which would cause a last zero column to appear in the matrix \(A=2{(x_h-x_1)}^{\top }\). Choosing \(h=1\) in this case would yield \(\Vert y\Vert ^2=d_{1,K+1}^2\), i.e., \(y_K=\pm \sqrt{d_{1,K+1}^2-\sum \limits _{j<K}y_j^2}\).
- 2.
In most of our papers, e.g., [84], we formulate this assumption in a more general way, stating that certain statements hold “with probability 1.” This is a reference to the fact that if d were uniformly sampled from a compact set, it would yield matrices A having full rank with probability 1.
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Liberti, L., Lavor, C. (2017). Realizing complete graphs. In: Euclidean Distance Geometry. Springer Undergraduate Texts in Mathematics and Technology. Springer, Cham. https://doi.org/10.1007/978-3-319-60792-4_3
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DOI: https://doi.org/10.1007/978-3-319-60792-4_3
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