Euclidean distance embedding

Ivanduc, O., Ivanduc, T. and Klein, D.J. (2001b) Intrinsic graph distances compared to Euclidean distances for correspondent graph embedding. MATCH Commun. Math. Comput. Chem., 44, 251-278. 

This subsection is devoted to the Sierpinski gasket d = 2) and its corresponding sponge d = 3), further on called Sierpinski triangular lattices. This fractal is characterized by a mass fractal dimension ds = ln(d -I- l)/ln2, which depends on the embedding spatial dimension d (see Fig. 2 for examples in d = 2 and d = 3). Note that for Sierpinski lattices in general, the Euclidean distance r between two lattice sites scales as the topological distance , i r, so that there is only one mass fractal dimension ds, M. ... 

MVU differs from other spectral techniques in that rather than constructing a feature matrix from measurable properties (i.e. covariance, Euclidean distance), it directly learns the feature matrix by solving a convex optimisation problem. Once the feature matrix has been learnt however, MVU fits in with other spectral techniques as the low-dimensional embedding is given as the top eigenvectors of Eq.(2.1). 

The simplest formulation of the packing problem is to give some collection of distance constraints and to calculate these coordinates in ordinary three-dimensional Euclidean space for the atoms of a molecule. This embedding problem - the Fundamental Problem of Distance Geometry - has been proven to be NP-hard [116]. However, this does not mean that practical algorithms for its solution do not exist [117-119]. 

Three-dimensional electron densities have no boundaries they converge to zero exponentially with distance from the nuclei of the peripheral atoms in the molecule. Considering a single, isolated molecule, the exact quantum-mechanical electron density becomes zero in a strict sense only at infinite distance from the center of mass of the molecule. Consequently, the electron density is not a compact set, just as the embedding three-dimensional Euclidean space E3 is not compact either. However, the three-dimensional Euclidean space E3, as a subset of a four-dimensional Euclidean space E4, can be slightly extended (for example, by adding one point) and made compact by various compactification techniques. 

An important question is whether the proximity measures are compatible with those of these references addresses the important issue of whether the proximity measure is compatible with embedding in a Euclidean space. For example, satisfying the distance axioms does not in itself guarantee that any distance matrix associated with a given set of molecules be compatible as the distance axioms are still satisfied in a non-Euclidean space. Gower has written extensively on this important issue, and his work should be consulted for details (89-91). Benigni (92), and Carbo (67) have also contributed interesting approaches in this area. 

The embedded cluster graph is obtained by making connections between the protein spots that are separated by Euclidean geometrical distances shorter than or equal to a selected critical distance [Randic and Basak, 2002 Bajzer, Randic ef al., 2003]. 

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