Intrinsic dimension of concept lattices
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Geometric analysis is a very capable theory to understand the influence of the high dimensionality of the input data in machine learning (ML) and knowledge discovery (KD). With our approach we can assess how far the application of a specific KD/ML-algorithm to a concrete data set is prone to the curse of dimensionality. To this end we extend V. Pestov's axiomatic approach to the instrinsic dimension of data sets, based on the seminal work by M. Gromov on concentration phenomena, and provide an adaptable and computationally feasible model for studying observable geometric invariants associated to features that are natural to both the data and the learning procedure. In detail, we investigate data represented by formal contexts and give first theoretical as well as experimental insights into the intrinsic dimension of a concept lattice. Because of the correspondence between formal concepts and maximal cliques in graphs, applications to social network analysis are at hand.
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