Topology-preserving map

Rose, V. S., Croall, I. E, MacFie, H. J. H. An apphcation of unsupervised neural network methodology (Kohonen topology-preserving mapping) to QSAR analysis. Quant. Struct.-Act. Relat. 1991,10,6-15. 

For example, C could be the category of topological spaces, or abstract simplicial complexes, or posets, in which case the structure-preserving maps would respectively be continuous maps, or simplicial maps, or order-preserving maps (see Definition 10.3). 

Intuitively, one can think of the map g as the pullback map. It is important to remark that if tp is not injective, it may happen that dim g r]) > dim 77. Since g is an order-preserving map, the induced map between abstract simphcial complexes A g) Bd (Horn (G, K)) — Bd (Horn (T, K)) is simplicial and gives the corresponding map of topological spaces, which we denote by V k- It is important to notice that the map g does not always come from a cellular map. 

The topological error measure (t), in which the respective first and second BMUs of all the data samples, are determined If these are not adjacent on the map, then, this is considered an error. The total error is then normalized to a range from 0 to 1, where 0 means perfect topology preservation (Polzlbauer, 2004). The actual topological error of the trained SOM map is 0.05. 

Due to the Kohonen learning algorithm, the individual weight vectors in the Kohonen map are arranged and oriented in such a way that the structure of the input space, i.e. the topology is preserved as well as possible in the resulting... 

By induction on dim(S). If dim(S) - 0 the theorem is trivially true. Let s assume dim(S) > 0. Since the assertion Is purely topological with respect to S and flatness is preserved under base change, we may assume that S is reduced, irreducible and normal (normalization is a closed mapping and the upper semi-continuity Is equivalent to the fact that the sets... 

Using this algorithm, the network is able to map the data so that similar data vectors excite neurons that are very near each other, thereby preserving the topology of the original sample vectors. Visual inspection of the map allows the user to identify outliers and recognize areas where groups of similar samples have clustered. 

These ideas also generalize neatly to higher-order systems. A fixed point of an th-order system is hyperbolic if all the eigenvalues of the linearization lie off the imaginary axis, i.e., Re(Aj iO for / = ,. . ., . The important Hartman-Grobman theorem states that the local phase portrait near a hyperbolic fixed point is topologically equivalent to the phase portrait of the linearization in particular, the stability type of the fixed point is faithfully captured by the linearization. Here topologically equivalent means that there i s a homeomorphism (a continuous deformation with a continuous inverse) that maps one local phase portrait onto the other, such that trajectories map onto trajectories and the sense of time (the direction of the arrows) is preserved. 

The iterative training procedure adapts the network in a way that similar input objects are also situated close together on the topological map. The network s topological layer can be seen as a two-dimensional grid, which is folded and distorted into the -dimensional input space to preserve the original structure as well as possible. Clearly, any attempt to represent an n-dimensional space in two dimensions will result in loss of detail however, the technique is useful to visualize data that might otherwise be hard to understand. 

If Y has finite limits and / preserves finite limits, then is exact by Lemma 2.7. It follows that a continuous map between topological spaces induces an admissible continuous functor between the corresponding sites. 

When the set of cells of a CW complex is given by means of a combinatorial enumeration, and the cell attachment maps are not too complicated, for instance if the CW complex in question is regular, it is natural to attempt to use the standard notion of cellular collapse to simplify the considered topological space, while preserving its homotopy t3rpe. 

A method to project atomistically detailed chains to smoother paths was proposed by Kroger ef al. [84] through a projection operation that maps a set r , i = 1,2,..., N, of N atomistic coordinates of a linear discrete chain to a new set R of N coarsegrained ones defining a smoother path for the chain that avoids the kinks of the original chain but preserves somewhat its topology (the main chain contour). 

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