Protein topology representations such as residue contact maps are an important intermediate step towards ab initio prediction of protein structure. Although improvements have occurred over the last years, the problem of accurately predicting residue contact maps from primary sequences is still largely unsolved. Among the reasons for this are the unbalanced nature of the problem (with far fewer examples of contacts than non-contacts), the formidable challenge of capturing long-range interactions in the maps, the intrinsic difficulty of mapping one-dimensional input sequences into two-dimensional output maps. In order to alleviate these problems and achieve improved contact map predictions, in this paper we split the task into two stages: the prediction of a maps principal eigenvector (PE) from the primary sequence, the reconstruction of the contact map from the PE and primary sequence. Predicting the PE from the primary sequence consists in mapping a vector into a vector. This task is less complex than mapping vectors directly into two-dimensional matrices since the size of the problem is drastically reduced and so is the scale length of interactions that need to be learned.