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Dimension reduction of data sets is a standard problem in the realm ofmachine learning and knowledge reasoning. They affect patterns in anddependencies on data dimensions and ultimately influence any decision-makingprocesses. Therefore, a wide variety of reduction procedures are in use, eachpursuing different objectives. A so far not considered criterion is theconceptual continuity of the reduction mapping, i.e., the preservation of theconceptual structure with respect to the original data set. Based on the notionscale-measure from formal concept analysis we present in this work a) thetheoretical foundations to detect and quantify conceptual errors in datascalings; b) an experimental investigation of our approach on eleven data setsthat were respectively treated with a variant of non-negative matrixfactorization.

Quantifying the Conceptual Error in Dimensionality Reduction