Most machine learning techniques are designed for vectorial data and compute distances or dot products between vectors. However, for many problems relational representations like matrices, trees, or graphs are more appropriate than vector-based descriptions. I will introduce a learning machine, which replaces the computed relations (distances or dot products) by the observed or measured relations between objects. Standard algorithms are not able to directly handle the measured relations because requirements like positive definiteness of the relation matrix cannot be guaranteed. The mathematical foundation of my approach is that measurements can -- under mild assumptions -- be expressed through a Hilbert-Schmidt-Operator and, therefore, model selection in the range of this operator is possible. My new approach excels in extracting relevant features from large data sets like world wide web linking matrices or gene expression profiles obtained from the microarray technique. On gene expression data sets from patients with cancer, where the outcome of a chemo- or radiation therapy must be predicted, I demonstrate that my method is superior to state of the art approaches. The improvement in prediction results from the extraction of important genes which are indicative for the therapy outcome. The improved predictive ability arises from the relational technique's extraction of the specific genes that indicate the likely therapy outcome.