  Data transformation, e.g. feature transformation and selection, is an
integral part of any machine learning procedure. In this paper we introduce an
information-theoretic model and tools to assess the quality of data
transformations in machine learning tasks. In an unsupervised fashion, we
analyze the transfer of information of the transformation of a discrete,
multivariate source of information X into a discrete, multivariate sink of
information Y related by a distribution PXY . The first contribution is a
decomposition of the maximal potential entropy of (X, Y) that we call a balance
equation, into its a) non-transferable, b) transferable but not transferred and
c) transferred parts. Such balance equations can be represented in (de Finetti)
entropy diagrams, our second set of contributions. The most important of these,
the aggregate Channel Multivariate Entropy Triangle is a visual exploratory
tool to assess the effectiveness of multivariate data transformations in
transferring information from input to output variables. We also show how these
decomposition and balance equation also apply to the entropies of X and Y
respectively and generate entropy triangles for them. As an example, we present
the application of these tools to the assessment of information transfer
efficiency for PCA and ICA as unsupervised feature transformation and selection
procedures in supervised classification tasks.
