Active Positive-Definite Matrix Completion.

In many applications, e.g., recommender systems and biological data analysis, the datasets of interest are positive definite (PD) matrices. Such matrices are usually similarity matrices, obtained by the multiplication of a matrix of preferences or observations with its transpose. Oftentimes, such real-world matrices are missing many entries and a fundamental data-analysis task, known by the term PD-matrix completion, is the inference of these missing entries. In this paper, we introduce the active version of PD-matrix completion, in which we assume access to an oracle that, at a given cost, returns the value of an unobserved entry of the PD matrix. In this setting, we consider the following question: “given a fixed budget, which entries should we query so that the completion of the new matrix is much more indicative of the underlying data?”. The main contribution of the paper is the formalization of the above question as the ActivePDCompletion problem and the design of novel and effective algorithms for solving it in practice.