  In this work we suggest a statistical mechanics approach to the
classification of high-dimensional data according to a binary label. We propose
an algorithm whose aim is twofold: First it learns a classifier from a
relatively small number of data, second it extracts a sparse signature, {\it
i.e.} a lower-dimensional subspace carrying the information needed for the
classification. In particular the second part of the task is NP-hard, therefore
we propose a statistical-mechanics based message-passing approach. The
resulting algorithm is firstly tested on artificial data to prove its validity,
but also to elucidate possible limitations.
  As an important application, we consider the classification of
gene-expression data measured in various types of cancer tissues. We find that,
despite the currently low quantity and quality of available data (the number of
available samples is much smaller than the number of measured genes, limiting
thus strongly the predictive capacities), the algorithm performs slightly
better than many state-of-the-art approaches in bioinformatics.
