Abstract
This paper deals with problems which require the reconstruction of structures of multi-dimensional dependencies from data. From a mathematical point of view these problems belong to the most complicated problems of artificial intelligence, such as reconstruction of the structures of multi-dimensional regressions or difference equations. In classification we meet similar problems when we have to select the space of classification features. Here we consider a special problem of supervisor-based classification that can be solved by the classification method “Alpha-procedure”. This problem consists in the following: Normally, the construction of the separating rule is performed during a training phase, where a supervisor defines the belonging of objects to classes by using a training set of data that can be small. But obviously the rich (partly subconscious) experience of the supervisor which is not described quantitatively somehow influences his decision. This may concern the importance, the uselessness, or even the harmfulness of the features. By this reason, the construction of the separation rule directly in the Euclidian data space leads to instability of that rule in certain cases. The paper explains why the Alpha-procedure, that performs an inductive construction of the separating rule in the homogeneous Lorentz space, allows a stable classification of new objects in the application phase without supervisor. It also shows, from the point of view of group transformations and their invariants, the difference between the mathematical apparatus for the search of the decision rule in a fixed feature space and in a space that is constructed by selecting features.
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Notes
- 1.
In the sense of the minimum of the intersection of objects belonging to different classes
- 2.
see footnote 1
- 3.
The totality of all lines and (two-dimensional) planes of a space which intersect a given point \( S \) of the space is called projective bundle of the lines and planes with the centre \( S \).
- 4.
The coefficient \( c = c_{1} = c_{2} = \ldots = c_{n} \) may be the speed of the light (taken with “minus”) in the special theory of relativity.
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Lange, T. (2020). Homogeneous Space as Media for the Inductive Selection of Separating Features for the Construction of Classification Rules. In: Shakhovska, N., Medykovskyy, M.O. (eds) Advances in Intelligent Systems and Computing IV. CSIT 2019. Advances in Intelligent Systems and Computing, vol 1080. Springer, Cham. https://doi.org/10.1007/978-3-030-33695-0_34
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