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We investigate dimensionality reduction methods from the perspective of their ability to produce a low-rank customer-product matrix representation. We analyze the results of using collaborative filtering based on SVD, RI, Reflective Random Indexing (RRI) and Randomized Singular Value Decomposition (RSVD) from the perspective of selected algebraic (i.e. application-independent) properties. We show that the Frobenius-norm optimality of SVD does not correspond to the optimal recommendation accuracy, when measured in terms of F1. On the other hand, a high collaborative filtering quality is achievable when a matrix decomposition-based on a combination of RRI and SVD referred to as RSVD-RRI-leads to increased diversity of low-dimensional eigenvectors. The diversity is observable from the perspective of cosine similarities analyzed in comparison to the analogical case of SVD. Such a feature is more desirable than the fidelity of the input matrix spectrum representation, despite the MSE-optimality of SVD.

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Widely-referenced approaches to collaborative filtering (CF) are based on the use of an input matrix that represents each user profile as a vector in a space of items and each item as a vector in a space of users. When the behavioral input data have the form of (userX, likes, itemY) and (userX, dislikes, itemY) triples one has to propose a representation of the user feedback data that is more suitable for the use of propositional data than the ordinary user-item ratings matrix. We propose to use an element-fact matrix, in which columns represent RDF-like behavioral data triples and rows represent users, items, and relations. By following such a triple-based approach to the bi-relational behavioral data representation we are able to improve the quality of collaborative filtering. One of the key findings of the research presented in this paper is that the proposed bi-relational behavioral data representation, while combined with reflective matrix processing, significantly outperforms state-of-the-art collaborative filtering methods based on the use of a ‘standard’ user-item matrix.

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